Multiplication of Polynomials
In this chapter, we are going to look at how to multiply polynomials. The good news is that we've already started to do this without even knowing it when we learnt how to expand binomials using the distributive law! Remember,
Not all questions are the same as binomial expansion though. We may be asked to evaluate the product of two polynomials.
For example, if $P$P$\left(x\right)=12$(x)=12 and $Q$Q$\left(x\right)=9$(x)=9, the product of these polynomials is $12\times9$12×9, which equals $108$108. ie. $P$P$\left(x\right)$(x)$Q$Q$\left(x\right)=108$(x)=108.
Or we may be asked to find missing terms in a polynomial (we'll see how in example video 3 below).
Every term in one pair of brackets has to be multiplied by every other term in all the other pairs of brackets.
Examples
Question 1
Expand $\left(a+2\right)\left(5a^2-2a+2\right)$(a+2)(5a2−2a+2).
Question 2
Given that $P($P($4$4$)$) $=$= $10$10 and $Q($Q($4$4$)$) $=$= $7$7, evaluate $P($P($4$4$)$)$Q($Q($4$4$)$).
Question 3
Consider the expansion of $\left(4x^3-4x^2+ax+b\right)\left(4x^2-2x+9\right)$(4x3−4x2+ax+b)(4x2−2x+9).
In the expansion, the coefficient of $x^3$x3 is $12$12. Form an equation and solve for the value of $a$a.
In the expansion, the coefficient of $x^2$x2 is $4$4. Form an equation and solve for the value of $b$b.