Problem 71 Factor $P(x)$ into linear fact... [FREE SOLUTION]

91影视

A Graphical Approach to College Algebra

John Hornsby,Margaret L. Lial,Gary Rockswold

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 71

Factor $P(x)$ into linear factors given that $k$ is a zero of $P$. $$P(x)=2 x^{3}-3 x^{2}-17 x+30 ; \quad k=2$$

Short Answer

Expert verified

$P(x) = (x - 2)(2x - 5)(x + 3)$.

Step by step solution

Verify that k is a zero

Evaluate the polynomial $P(x)$ at $x = k$ to verify that $k = 2$ is indeed a zero. Substitute $x = 2$ into the polynomial: \[P(2) = 2(2)^3 - 3(2)^2 - 17(2) + 30 \] Simplify it step by step: \[= 2(8) - 3(4) - 34 + 30 \]\[= 16 - 12 - 34 + 30 \]\[= 0\] Since $P(2) = 0$, $k = 2$ is a zero of $P(x)$.

Synthetic Division

Use synthetic division to divide $P(x)$ by $x - 2$, since $k = 2$ is a zero.The coefficients of $P(x) = 2x^3 - 3x^2 - 17x + 30$ are: 2, -3, -17, 30.Set up the synthetic division with 2 outside:\[\begin{array}{r|rrrr}2 & 2 & -3 & -17 & 30 \ & & 4 & 2 & -30 \ \hline & 2 & 1 & -15 & 0 \\end{array}\]The bottom row gives the quotient: $2x^2 + x - 15$ and remainder 0, confirming that $x - 2$ divides $P(x)$ exactly.

Factor the Quadratic

Now factor the resulting quadratic $2x^2 + x - 15$. We need two numbers that multiply to $2 \times -15 = -30$ and add to $1$ (the coefficient of $x$). These numbers are 6 and -5.Rewrite and factor by grouping:\[2x^2 + 6x - 5x - 15 \]Group the terms:\[(2x^2 + 6x) + (-5x - 15)\]Factor each group:\[2x(x + 3) - 5(x + 3)\]Factor out the common factor $(x + 3)$:\[(2x - 5)(x + 3)\]

Combine Factors

Combine the linear factor found from synthetic division $(x - 2)$ with the factors of the quadratic $(2x - 5)(x + 3)$ to express $P(x)$ fully in linear factors:$P(x) = (x - 2)(2x - 5)(x + 3)$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Synthetic Division Simplified

Synthetic division is a quick method for dividing polynomials, especially when you're dividing by a linear factor like $x - k$. It's simpler and faster than long division, making it a preferred technique in many scenarios.
Here鈥檚 how to apply synthetic division:

Write down the coefficients of the polynomial you wish to divide, in this case, $2x^3 - 3x^2 - 17x + 30$.
Place the zero of the divisor, $k = 2$, outside of the division symbol.
Bring down the leading coefficient, which is $2$ in this problem.
Multiply $2$ by $k$, add it to the next coefficient to get a new row entry, and repeat this process.

For our example, synthetic division showed that $x - 2$ is a divisor of $P(x)$, yielding a quotient of $2x^2 + x - 15$. This step confirmed $x - 2$ evenly divides $P(x)$ with a remainder of zero.

Understanding Zero of Polynomial

The zero of a polynomial refers to a solution of the equation $P(x) = 0$. Finding this number, denoted as $k$, is crucial when factoring polynomials.
To determine if $k$ is a zero of $P(x)$:

Substitute $k$ into the polynomial.
Calculate $P(k)$.

If $P(k) = 0$, then $k$ is indeed a zero. In our example, substituting $k = 2$ into $P(x)$ yields zero. This result confirms $2$ is a zero of the polynomial $P(x) = 2x^3 - 3x^2 - 17x + 30$. Knowing the zero allows us to perform synthetic division, simplifying the polynomial for further factorization.

Quadratic Factorization Unlocked

When you reduce a cubic polynomial via synthetic division, often you're left with a quadratic polynomial to factor. Here, after division, we have $2x^2 + x - 15$. Factoring a quadratic requires finding two numbers that:

Multiply to the product of the leading coefficient and the constant term (in this case $2 \times -15 = -30$).
Add up to the middle term's coefficient (here, $1$).

For the quadratic $2x^2 + x - 15$, those numbers are $6$ and $-5$. Rewrite the middle term using these numbers and factor by grouping:

Group into $(2x^2 + 6x)$ and $(-5x - 15)$.
Factor out common factors: $2x(x + 3) - 5(x + 3)$.
The common binomial $(x + 3)$ appears, allowing to factor into $(2x - 5)(x + 3)$.

Together with the linear factor obtained earlier, the fully factored form of the polynomial becomes $(x - 2)(2x - 5)(x + 3)$.

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91影视

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=2 x^{3}-3 x^{2}-17 x+30 ; \quad k=2$$

Short Answer

Step by step solution

Verify that k is a zero

Synthetic Division

Factor the Quadratic

Combine Factors

Key Concepts

Synthetic Division Simplified

Understanding Zero of Polynomial

Quadratic Factorization Unlocked

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