Determine for which polynomials (x+2) is a factor. Explain your answer. P(x)=x^4-3x^3-16x-12 Q(x)=x^3-3x^2-16x-12

Jaya Legge 2021-09-15 Answered

Determine for which polynomials (x+2) is a factor. Explain your answer.
\(\displaystyle{P}{\left({x}\right)}={x}^{{4}}-{3}{x}^{{3}}-{16}{x}-{12}\)

\(\displaystyle{Q}{\left({x}\right)}={x}^{{3}}-{3}{x}^{{2}}-{16}{x}-{12}\)

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Expert Answer

averes8
Answered 2021-09-16 Author has 15302 answers

We have to show if (x+2) is a factor of the polynomials \(\displaystyle{P}{\left({x}\right)}={x}^{{4}}-{3}{x}^{{3}}-{16}{x}-{12}\) and \(\displaystyle{Q}{\left({x}\right)}={x}^{{3}}-{3}{x}^{{2}}-{16}{x}-{12}\) or not.
To check the factor we have to see whether (x+2) satisfies the given polynomials.
If it satisfies the given polynomial or gives the answer 0 i.e P(x)=0 or Q(x)=0 we will say that it is a factor of the polynomial, if it doesn't satisfy the given polynomial we will say that it is not a factor of the polynomial.
First, we will put \(\displaystyle{x}+{2}={0}\Rightarrow{x}=-{2}\)
Now, we will substitute x=-2 in the given polynomials P(x) and Q(x) and see if it is a factor of these polynomials or not.
Now,
\(\displaystyle{P}{\left(-{2}\right)}={\left(-{2}\right)}^{{4}}-{3}{\left(-{2}\right)}^{{3}}-{16}{\left(-{2}\right)}-{12}\)
\(\displaystyle{P}{\left(-{2}\right)}={16}+{24}+{32}-{12}\)
\(\displaystyle{P}{\left(-{2}\right)}={60}\)
therefore, \(\displaystyle{P}{\left(-{2}\right)}\ne{0}\)
hence, \(x=-2\) is not a factor of the polynomial P(x)
Now,
\(\displaystyle{Q}{\left(-{2}\right)}={\left(-{2}\right)}^{{3}}-{3}{\left(-{2}\right)}^{{2}}-{16}{\left(-{2}\right)}-{12}\)
\(\displaystyle{Q}{\left(-{2}\right)}=-{8}-{12}+{32}-{12}\)
\(\displaystyle{Q}{\left(-{2}\right)}={0}\)
therefore, \(Q(-2)=0\)
Hence, \(x=-2\) is a factor of the polynomial \(Q(x)\)

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