Multiplying Polynomials

Summary:

Polynomials can be multiplied using the distributive property multiply times. To do this, choose one polynomial (it's generally best to go with the shorter one), then separately distribute each term to the other polynomial and add them up.

For these problems, it is best to be methodical. Make sure to write out every term after the multiplication, as adding terms together in your head is guaranteed to lead to silly mistakes.

Example 1:

Evaluate the following:
\((2x+5)(3x-4)\)

We will choose the first binomial in this case, since it doesn't contain a negative term. First, we will split it up into the following: $$2x(3x-4) + 5(3x-4)$$ We can now distribute the \(2x\) and the \(5\) $$6x^2-8x+15x-20$$ And finally we can simplify this by combining like terms to get $$6x^2 + 7x -20$$

You can also multiply polynomials with more than two terms, which you will see in the next example

Example 2:

Evaluate the following:
\((2x-3)(4x^2-7x+3)\)

We will choose the binomial \(2x+3\), since it has less terms. Just like in the last problem, we will split it up: $$2x(4x^2-7x+3) - 3(4x^2-7x+3)$$ We can now distribute each term individually to get $$8x^3-14x^2+6x - 12x^2-(-21x) - 9$$ Next, we can simplify like terms to get $$8x^3 - 26x^2 + 27x -9 $$

You can also multiply more than two polynomials, as we will demonstrate in the next example

Example 3:

Evaluate the following:
\((3x-2)(4x+1)(x+2)\)

First we will multiply \(3x-2\) and \(4x+1\) as normal, ignoring the third term for now. To do that, we will split up the first term as we did in the previous two problems: $$(3x(4x+1) - 2(4x+1))(x+2)$$ We can then distribute the \(3x\) and the \(-2\) to get $$(12x^2+3x - 8x-2)(x+2) = (12x^2-5x-2)(x+2)$$ We now need to multiply again, this time choosing \(x+2\) to split $$x(12x^2-5x-2)+2(12x^2-5x-2)$$ We can now distribute again to get $$12x^3-5x^2-2x+24x^2-10x-4$$ Finally, we can simplify this to get the answer $$12x^3 +19x^2-8x+4$$

Conlcusion:
In conclusion you learned how to multiply polynomials. You will eventually learn how to reverse this by facotring them.

Next: Using Tables on your calculator