Factor higher-degree polynomials

Factoring higher-degree polynomials involves using structure—common factors, grouping, identities, substitution, the Factor Theorem, the Rational Root Theorem, and polynomial division—to express a polynomial as a product of lower-degree factors. The factorization connects factors with zeros, multiplicities, and graph behavior, while preserving equivalence through expansion; work is limited to polynomial degrees and integer or rational coefficients typically encountered, without requiring general formulas or factorization over abstract number systems.

Detailed Explanation: Factor higher-degree polynomials

When a polynomial has degree $3$ or greater, look for rational zeros first. If $r$ is a zero of $P(x)$ , then $(x-r)$ is a factor.

Consider

P(x)=x^4-2x^3-7x^2+8x+12.

1. List possible rational zeros

Because the leading coefficient is $1$ , the Rational Root Theorem says that any rational zero must be a factor of the constant term $12$ :

\pm1,\ \pm2,\ \pm3,\ \pm4,\ \pm6,\ \pm12.

Test a simple candidate:

P(2)=16-16-28+16+12=0.

Therefore, $x=2$ is a zero, so $(x-2)$ is a factor.

2. Divide by the factor

Use synthetic division:

Thus,

P(x)=(x-2)(x^3-7x-6).

3. Factor the remaining polynomial

Test another possible rational zero in the cubic:

3^3-7(3)-6=27-21-6=0.

So $x=3$ is a zero, and $(x-3)$ is a factor. Dividing gives

x^3-7x-6=(x-3)(x^2+3x+2).

Now factor the quadratic:

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

Therefore, the complete factorization is

\boxed{x^4-2x^3-7x^2+8x+12=(x-2)(x-3)(x+1)(x+2)}.

Always check by multiplying the factors back together. The product expands to the original polynomial, confirming that the factorization is correct.

Learn by doing: Factor higher-degree polynomials

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Factor Polynomials (Order 4) - As Quadratic (No Hint), Coefficient N

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