Answer to Question #240907 in Calculus for CCW

Question #240907

Factoring Polynomials Practice:

Use synthetic division to divide polynomials by possible roots


x3 + 4x2 + 5x + 2

P= Q=


1
Expert's answer
2021-09-24T16:35:03-0400

Since all coefficients are integers, we can apply the rational zeros theorem.

The trailing coefficient (the coefficient of the constant term) is 2.2.

Find its factors (with the plus sign and the minus sign): ±1,±2.±1,±2.

These are the possible values for p.p.

The leading coefficient (the coefficient of the term with the highest degree) is 1.1. Find its factors (with the plus sign and the minus sign): ±1.±1.

These are the possible values for q.q.

Find all possible values of pq:±11,±21.\dfrac{p}{q}:\pm\dfrac{1}{1}, \pm\dfrac{2}{1} .

Simplify and remove the duplicates (if any).

These are the possible rational roots: ±1,±2.\pm1, \pm2.

Next, check the possible roots: if aa is a root of the polynomial P(x),P(x), the remainder from the division of P(x)P(x) by xax−a should equal 0.0.

Check 1:1: divide x3+4x2+5x+2x^3 + 4x^2 + 5x + 2 by x1.x-1.

P(1)=1+4+5+2=12;P(1)=1+4+5+2=12;  thus, the remainder is 12.12.


Check 1:-1: divide x3+4x2+5x+2x^3 + 4x^2 + 5x + 2 by x+1.x+1.

P(1)=1+45+2=0;P(-1)=-1+4-5+2=0;  thus, the remainder is 0.0.

Hence, 1−1 is a root.


Check 2:2: divide x3+4x2+5x+2x^3 + 4x^2 + 5x + 2 by x2.x-2.

P(2)=8+16+10+2=36;P(2)=8+16+10+2=36;  thus, the remainder is 36.36.


Check 2:-2: divide x3+4x2+5x+2x^3 + 4x^2 + 5x + 2 by x+2.x+2.

P(2)=8+1610+2=0;P(-2)=-8+16-10+2=0;  thus, the remainder is 0.0.

Hence, 2−2 is a root.



x3x2x1x0114521321320\def\arraystretch{1.5} \begin{array}{c:c} & x^3 & x^2 & x^1 & x^0 \\ -1 & 1 & 4 & 5 & 2 \\ & & -1 & -3 & -2\\\hline & 1 & 3 & 2 & 0 \\ \end{array}

x3+4x2+5x+2=(x+1)(x2+3x+2)x^3+4x^2+5x+2=(x+1)(x^2+3x+2)



x2x1x0213222110\def\arraystretch{1.5} \begin{array}{c:c} & x^2 & x^1 & x^0 \\ -2 & 1 & 3 & 2 \\ & & -2 & -2 \\\hline & 1 & 1 & 0 \\ \end{array}

x2+3x+2=(x+2)(x+1)x^2+3x+2=(x+2)(x+1)

Then


x3+4x2+5x+2=(x+1)2(x+2)x^3+4x^2+5x+2=(x+1)^2(x+2)

Roots are: 2-2 of multiplicity 1,11, -1 of multiplicity 2.2.

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