Question

which expression represents the product of $x^{3}+2x - 1$ and $x^{4}-x^{2}+3$?
$x^{12}-x^{8}+x^{4}+2x^{5}+6x - 3$
$x^{7}-2x^{3}-3$
$x^{7}-x^{5}+2x^{3}-3x^{4}+4x^{3}+6x - 3$
$x^{4}+2x + 2$

Explanation:

Step1: Multiply each term of first polynomial by second

\((x^{3}+2x - 1)(x^{4}-x^{2}+3)\)
\(=x^{3}(x^{4}-x^{2}+3)+2x(x^{4}-x^{2}+3)-1(x^{4}-x^{2}+3)\)

Step2: Apply distributive property

\(=x^{3}\cdot x^{4}-x^{3}\cdot x^{2}+3x^{3}+2x\cdot x^{4}-2x\cdot x^{2}+6x - x^{4}+x^{2}-3\)

Step3: Use exponent rule \(a^{m}\cdot a^{n}=a^{m + n}\)

\(=x^{3 + 4}-x^{3+2}+3x^{3}+2x^{1 + 4}-2x^{1+2}+6x - x^{4}+x^{2}-3\)
\(=x^{7}-x^{5}+3x^{3}+2x^{5}-2x^{3}+6x - x^{4}+x^{2}-3\)

Step4: Combine like - terms

\(=x^{7}+(-x^{5}+2x^{5})+(3x^{3}-2x^{3})-x^{4}+x^{2}+6x - 3\)
\(=x^{7}+x^{5}+x^{3}-x^{4}+x^{2}+6x - 3\)

Answer:

None of the given options are correct.