Question

Multiplying Polynomials and Simplifying Expressions

Answer 1

Given:

Polynomials: $a+3 \text { and }-2 a^{2}+15 a+6 b^{2}$

To find:

The product of the polynomials.

Solution:

$(a+3)(-2 a^{2}+15 a+6 b^{2})$

Using distributive property: $x(y+z)=xy+xz$

$(a+3)(-2 a^{2}+15 a+6 b^{2})=a(-2 a^{2}+15 a+6 b^{2})+3(-2 a^{2}+15 a+6 b^{2})$

Now multiply each of the first term with each of the second term.

                             $=a\left(-2 a^{2}\right)+a \cdot 15 a+a \cdot 6 b^{2}+3\left(-2 a^{2}\right)+3 \cdot 15 a+3 \cdot 6 b^{2}$

Applying plus minus rule: $+(-x)=-x$

                            $=-2 a^{2} \cdot a+15 a \cdot a+6 a\cdot b^{2}-3 \cdot 2 a^{2}+3 \cdot 15 a+3 \cdot 6 b^{2}$

Apply the exponent rule: $x^{n} \cdot x^{m}=x^{n+m}$

                          $=-2 a^{3}+15 a^2+6 a b^{2}-6 a^{2}+45 a+18 b^{2}$

Add or subtract the like terms:

                          $=-2 a^{3}+15 a^2-6a^2+6 a b^{2}+45 a+18 b^{2}$

                          $=-2 a^{3}+9 a^{2}+6 a b^{2}+45 a+18 b^{2}$

Arrange in the order.

                          $=-2 a^{3}+9 a^{2}+45 a+6 a b^{2}+18 b^{2}$

The product of $a+3 \text { and }-2 a^{2}+15 a+6 b^{2}$ $-2 a^{3}+9 a^{2}+45 a+6 a b^{2}+18 b^{2}$.