Divide the polynomials -13x^2+4x^3+2x-7 and explain each step. State the answer in full sentence and ecxpress it as one expression.

lwfrgin 2021-09-19 Answered
Divide the polynomials \(\displaystyle-{13}{x}^{{2}}+{4}{x}^{{3}}+{2}{x}-{7}\) and explain each step. State the answer in full sentence and ecxpress it as one expression. Check your solution.

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Expert Answer

Daphne Broadhurst
Answered 2021-09-20 Author has 5477 answers
Given:
To divide the polynomials \(\displaystyle{\frac{{{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}}}{{{x}^{{2}}+{3}{x}-{2}}}}\) and verify the solution.
To divide the polynomials, first find the quotient by dividing the higher degree of each polynomial.
\(\displaystyle{\frac{{{4}{x}^{{3}}}}{{{x}^{{2}}}}}={4}{x}\)
Let's divide the polynomial with quotient 4x.
\(\displaystyle{Q}={4}{x},{R}=-{25}{x}^{{2}}+{10}{x}-{7}\)
Thus it can be expressed as,
Now, apply it with the expressions obtained above.
\(\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={4}{x}-{15}{x}^{{2}}+{3}{x}-{2}+{\left(-{5}{x}\right)}-{37}\)
Let's prove the result using the single term expressed.
\(\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={\left({4}{x}-{25}\right)}{\left({x}^{{2}}+{3}{x}-{2}\right)}+{\left({85}{x}-{57}\right)}\)
\(\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={4}{x}^{{3}}+{12}{x}^{{2}}-{8}{x}-{25}{x}^{{2}}-{75}{x}+{50}+{85}{x}-{57}\)
\(\displaystyle{4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}={4}{x}^{{3}}-{13}{x}^{{2}}+{2}{x}-{7}\)
LHS=RHS
Hence answer is verified.
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