to view answer.The following expressions are examples of polynomials that are in factored form.
$$
\left( {3x + 2} \right)\left( {4x - 3} \right)
$$
$$
\left( {4x^2 - 2x + 7} \right)\left( {3x + 8} \right)
$$
$$
\left( {3x^2 + 9} \right)\left( {4x^3 - 5x + 7} \right)
$$
To “expand” these expressions, we simply multiply the polynomial factors together to get one “expanded” polynomial. In order to multiply the polynomials, we need to use the distributive property.
Multiply the following binomials:
\[
\left( {3 + x} \right)\left( {6 - y} \right)
\]
1. To begin, we use the distributive property to multiply each term of the binomial $$ \left( {3 + x} \right) $$ the entire binomial $$(6-y)$$. Here, we get:
\[
\eqalign{&\left( {3 + x} \right)\left( {6 - y} \right) \\ &= 3\cdot(6-y)+x\cdot(6-y)}
\]
2. Now, we use the distributive property twice more to multiply the monomial 3 by each term of the binomial $$(6-y)$$ and we also multiply the monomial x by each term of the binomial $$(6-y)$$.
\[
\eqalign{ &18 + 6x - 3y - xy \\ =&3\cdot(6-y)+x\cdot(6-y)\\ =&3(6)+3(-y)+x(6)+x(-y)}
\]
This is our final, “expanded” polynomial.
[In the past, you may have been taught FOIL, which works for multiplying two binomials together. Note here that the $$6 (3)$$ are the First terms, the $$3 (- y)$$ are the Outer terms, the $$x ( 6)$$ are the Inner terms, and the $$x(-y)$$ are the Last terms.]
Now, let's look at multiplying trinomials:
Multiply the following polynomials:
\[
\left( {3x^2 - xy + 2x} \right)\left( { - x^2 - 4x + 7} \right)
\]
1. To begin, we use the distributive property to multiply each term of the polynomial $$\left(3x^2 - xy + 2x \right)$$ by the entire polynomial $$(-x^2-4x+7)$$. Here, we get:
\[
3x^2(-x^2-4x+7)+(-xy)(-x^2-4x+7)+2x(-x^2-4x+7)
\]
NOTE: Be careful with negatives. Make sure that when distributing to a negative term, the term stays negative (put them in parentheses).
2. Now, use the distributive property to multiply each monomial by each term in the trinomial $$(3x^2-xy+2x)$$.
\[\eqalign{
&3x^2(-x^2-4x+7)+(-xy)(-x^2-4x+7)+2x(-x^2-4x+7)\\=&3x^2(-x^2)_3x^2(-4x)+3x^2(7)+(-xy)(-x^2)+(-xy)(-4x)+(-xy)(7) + 2x(-x^2)+2x(-4x)+2x(7)}
\]
Note: Notice that every term in each of the original two polynomials is multiplied together.
Once the terms are multipled, we obtain:
\[
-3x^4-12x^3+21x^2+x^3y+4x^2y-7xy-2x^3-8x^2+14x
\]
3. Finally, combine like terms to get the final “expanded” polynomial:
\[
- 3x^4 + x^3y - 14x^3 + 4x^2 y + 13x^2 - 7xy + 14x
\]
NOTE: While we typically write polynomials in descending order, it is also equivalent to write the polynomials in any order as long as we are careful to keep negatives and positives with the correct terms.
Check your Skills by completing the following problems:
Multiply the following polynomials (“Expand” the expressions):
1. $$
\left( {x - 4} \right)\left( {x + 3} \right)
$$
Think about it for a moment and then access this link to view answer.
2. $$
\left( {3 - 2x} \right)\left( {2x + 4} \right)
$$
Think about it for a moment and then access this link to view answer.
3. $$
\left( {3y - 2z} \right)\left( {5a + 7b^2 } \right)
$$
Think about it for a moment and then access this link to view answer.
4. $$(3x^2+7x-2)(6x^2-4x+7)
$$
Think about it for a moment and then access this link to view answer.
5. $$
\left( {13x^{12} - 7y^2 + 76} \right)\left( {12z^4 + 23} \right)
$$
Think about it for a moment and then access this link to view answer.