Lecture: Polynomials & Factoring
Lecture script · about 15–20 minutes spoken · Week 6
Opening
Picture a square garden bed behind a community center, x feet on each side. The volunteers stretch it 3 feet one way and 4 feet the other. Now its area is x plus 3, times x plus 4. A landscaping crew needs that area as one formula. A packaging designer needs the same kind of formula for a box folded from a flat sheet. A kinesiology student needs one for the height of a thrown ball. In every case, you build an expression by multiplying, and later you take it apart again.
Here's where we are in this course. In Week 1 you learned the distributive property and the exponent rules, so x times x is x squared. In Week 3 you evaluated expressions at a number, and that skill becomes your check in this lecture. In Weeks 4 and 5 you worked with lines and pairs of lines. Now the x squared terms arrive. Next week, in Week 7, you use what we build here to solve quadratic equations.
I'll take nine ideas in order: the names of the parts, adding and subtracting, multiplying, three special products, the greatest common factor, two kinds of trinomials, squares, and grouping. Every one of them ends the same way: you multiply back out to check.
1. Name the parts of a polynomial
Think about a grocery receipt. It's a list of items, each with an amount, added up. A polynomial is the same kind of list.
A polynomial is an expression built by adding terms, where each term is a number times a variable raised to a whole-number exponent. A term is one piece of that sum, separated from the next piece by a plus or minus sign. Take 3x squared minus 2x plus 5, which you evaluated back in Week 1. Its terms are 3x squared, negative 2x, and 5.
Three more names do most of the work. The degree is the largest exponent on the variable. Here it's 2. The leading coefficient is the number that multiplies the highest-power term. Here it's 3. The constant term is the term with no variable in it. Here it's 5.
Polynomials also get names by counting terms. A monomial has exactly one term, a binomial has exactly two, and a trinomial has exactly three. So 3x squared minus 2x plus 5 is a trinomial.
Pause and try this. Look at 6x cubed minus 9x squared. How many terms does it have, and what is its degree?
Count the pieces first, then find the biggest exponent.
Here's the answer. It has two terms, so it's a binomial, and its biggest exponent is 3, so its degree is 3, with leading coefficient 6.
2. Add and subtract: send the minus to every term
Imagine combining two fruit orders. You add crates of apples to crates of apples and bags of limes to bags of limes. You never add a crate to a bag. Polynomials work the same way: you combine like terms, the x squared terms together, the x terms together, the plain numbers together.
Adding x squared minus 6x plus 5 to x squared minus 2x minus 15 gives 2x squared minus 8x minus 10. Subtracting takes one extra move. A minus sign in front of parentheses multiplies every term inside by negative 1. That's the Week 1 distributive property again.
So x squared minus 6x plus 5, minus the whole of x squared minus 2x minus 15, becomes x squared minus 6x plus 5 minus x squared plus 2x plus 15. The x squared terms cancel. Negative 6x plus 2x is negative 4x. And 5 plus 15 is 20. The difference is negative 4x plus 20.
The classic slip is sending the minus to the first term only. That gives negative 8x minus 10, and it's wrong.
Pause and try this. In that subtraction, what does minus negative 15 turn into?
Say the sign out loud before you read on.
Here's the answer. Minus negative 15 becomes plus 15, because the minus multiplies every term, and two negatives make a positive.
3. Multiply with FOIL and check at a value
Back to the garden bed. To turn x plus 3, times x plus 4, into a sum, every term of the first factor has to multiply every term of the second. Two terms times two terms makes four products.
FOIL is a memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms, and the Last terms, then add the four products.
Let's do the week's shared polynomial: x minus 1, times x minus 5. First: x times x is x squared. Outer: x times negative 5 is negative 5x. Inner: negative 1 times x is negative x. Last: negative 1 times negative 5 is positive 5. Combine the middle two, and you get x squared minus 6x plus 5.
How do you know it's right? Pick a number and evaluate both forms. At x equals 2, the factored form gives 1 times negative 3, which is negative 3. The expanded form gives 4 minus 12 plus 5, which is also negative 3. They agree. In fact, the two forms agree at every input. From x equals 0 up to 6, both give 5, 0, negative 3, negative 4, negative 3, 0, and 5.
Pause and try this. The garden bed is x plus 3 by x plus 4. If the original side was 2 feet, how many square feet of soil does the bigger bed need?
Work out the two new side lengths first.
Here's the answer. The sides become 5 feet and 6 feet, and 5 times 6 is 30 square feet, which matches x squared plus 7x plus 12 at x equals 2.
4. Use the three special products
Here's a campus example. A square poster is x inches on a side, and it goes into a frame that adds 3 inches on every side. The framed square is x plus 6 on a side, so its area is x plus 6, squared.
Picture the framed square in four pieces: the poster itself, two strips that are x by 6, and a little 6 by 6 corner. That's x squared, plus 6x, plus 6x, plus 36. The two strips are the part people forget.
The special products are three multiplication patterns worth knowing by sight. The square of a sum is the first term squared, plus twice the product of the two terms, plus the second term squared. The square of a difference is the same, with a minus on the middle term. And a sum times a difference is the first squared minus the second squared. So x plus 6, squared, is x squared plus 12x plus 36. That's a perfect-square trinomial: a trinomial that is the square of a binomial.
The sum-times-difference pattern has a story too. A square patio gets 4 feet longer and 4 feet narrower. Its new area is x squared minus 16. The middle terms cancel, so the patio loses 16 square feet, whatever its size.
Pause and try this. Someone writes x plus 6, squared, as x squared plus 36. Using x equals 2, how do you show it's wrong?
Evaluate both sides at 2.
Here's the answer. The true value is 8 squared, which is 64, while x squared plus 36 gives only 40, so the middle term 12x is missing.
5. Factor out the greatest common factor first
Factoring runs multiplication backward. Picture three friends at a café, each ordering a 6-dollar salad and a 9-dollar entrée. The bill is 3 salads plus 3 entrées, or 3 times the 15-dollar meal. Pulling out that shared 3 is factoring.
Factoring is rewriting a polynomial as a product of simpler polynomials. The greatest common factor, or GCF, is the largest expression that divides every term. Take 6x cubed minus 9x squared. The numbers 6 and 9 share 3. The powers x cubed and x squared share x squared. So the GCF is 3x squared, and the factored form is 3x squared times 2x minus 3. Multiply it back, and you get the original.
The launched ball is a good second example. Its height after t seconds is negative 16 t squared plus 64t plus 80. Because the leading coefficient is negative, pull out negative 16. Inside, you're left with t squared minus 4t minus 5.
Always ask "what do all the terms share?" first. Skip it, and a problem like 3x squared minus 75 looks stuck.
Pause and try this. What is the greatest common factor of 3x squared and 75?
Look at the numbers only, since 75 has no x.
Here's the answer. It's 3, because 3 divides both 3 and 75, and 75 has no x, so no power of x can come out.
6. Factor trinomials that start with x squared
Now run FOIL backward. When you multiplied x plus 3 by x plus 4, the last numbers multiplied to 12, and the outer and inner products added to 7x. So to factor x squared plus 7x plus 12, you hunt for two numbers that multiply to 12 and add to 7. The pairs 1 and 12, 2 and 6, and 3 and 4 all multiply to 12. Only 3 and 4 add to 7. So it factors as x plus 3, times x plus 4.
Signs need care. For x squared minus 5x minus 14, you need a product of negative 14 and a sum of negative 5. Negative 7 and 2 work. For the shared polynomial, x squared minus 6x plus 5, you need a product of 5 and a sum of negative 6. Negative 1 and negative 5 work, and you're right back at x minus 1, times x minus 5. Multiplying and factoring are the same road, traveled in opposite directions.
Pause and try this. Someone factors x squared minus 5x minus 14 as x plus 7, times x minus 2. Why is that wrong?
Check the sum, with its signs.
Here's the answer. Positive 7 plus negative 2 is positive 5, not negative 5, so that version expands to x squared plus 5x minus 14.
7. Factor with a leading coefficient: the ac method
When the x squared term has a number in front, that number joins the hunt. Take 2x squared plus 7x plus 3. The ac method is a way to factor a trinomial when the leading coefficient is not 1. You find two numbers that multiply to a times c and add to b. You use them to split the middle term, and then you factor by grouping.
Here a is 2, b is 7, and c is 3, so a times c is 6. The numbers 1 and 6 multiply to 6 and add to 7. Split the 7x into x plus 6x. Now group: the first pair gives x times 2x plus 1, and the second pair gives 3 times 2x plus 1. Both pieces share 2x plus 1, so the result is 2x plus 1, times x plus 3. Multiply it back, and you get 2x squared plus 7x plus 3.
Pause and try this. Why won't a hunt for two numbers that multiply to 3 and add to 7 work here?
Think about the 2 in front of x squared.
Here's the answer. That hunt finds no pair, because the leading 2 takes part in every product, so you have to multiply a and c first.
8. Spot a difference of squares, and leave a sum of squares alone
Remember the patio, where x plus 4 times x minus 4 collapsed to x squared minus 16. Run that backward. A difference of squares is a binomial of the form a squared minus b squared, and it always factors as a sum times a difference. So 9x squared minus 25, which is the square of 3x minus the square of 5, factors as 3x plus 5, times 3x minus 5. And 3x squared minus 75 finally gets unstuck: pull out 3 to get 3 times x squared minus 25, then factor that as x minus 5, times x plus 5.
Now flip the sign. A sum of squares like x squared plus 16 does not factor over the real numbers. Try x plus 4 times x plus 4, and you get x squared plus 8x plus 16. Try x plus 4 times x minus 4, and you get x squared minus 16. Neither one works. That makes x squared plus 16 a prime polynomial. Here's an honest second reason: x squared is never negative, so x squared plus 16 is always at least 16 and never reaches zero.
Pause and try this. Which one factors over the real numbers: x squared minus 16, or x squared plus 16?
Decide before you read on.
Here's the answer. Only x squared minus 16 factors, as x plus 4 times x minus 4, because a sum of squares is prime over the real numbers.
9. Factor four terms by grouping, then check by expanding
Some polynomials have four terms and no single pattern. Factoring by grouping is a way to factor four terms: pair them, pull the greatest common factor out of each pair, and then pull out the binomial the two pairs share.
Take x cubed plus 2x squared plus 3x plus 6. The first pair, x cubed plus 2x squared, gives x squared times x plus 2. The second pair, 3x plus 6, gives 3 times x plus 2. Both share x plus 2, so the result is x plus 2, times x squared plus 3. Multiply it back: x cubed plus 3x plus 2x squared plus 6. That's the original, rearranged.
Here's the habit that ties the whole lecture together. Every factoring gets checked by expanding. If the product doesn't match, the factoring is wrong.
Pause and try this. Is x squared times x plus 2, plus 3 times x plus 2, a finished factoring?
Ask whether it's one product or a sum of two pieces.
Here's the answer. Not yet, because it's still a sum of two pieces; pulling out the shared x plus 2 turns it into one product.
Closing
Let's pull it together. A polynomial is a sum of terms, with a degree and a leading coefficient you can read off. To subtract, send the minus to every term. FOIL makes four products, and a check at one number tells you whether you got them right. The square of a binomial has a middle term, so x plus 6, squared, is never x squared plus 36. Factor out the GCF first. For x squared plus b x plus c, find a product of c and a sum of b; with a leading coefficient, use the ac method. A difference of squares factors, a sum of squares doesn't, and four terms can often be grouped. Then expand to check.
Next in this module come the slide deck, the Readings & Resources page, the lecture tutorial, the practice set, the quiz, and the discussion. Next week, in Quadratic Equations, those factored forms become a method for solving equations like x squared minus 6x plus 5 equals 0.
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