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Chapter 4: Problem 40

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=12 x^{3}-20 x^{2}+x+3 $$

Short Answer

Expert verified
The rational zeros are \(\frac{1}{2}\), and \(P(x)\) in factored form is \((2x - 1)(3x + 1)(2x - 3)\).

Step by step solution

01

Apply the Rational Root Theorem

The Rational Root Theorem states that any rational zero of a polynomial, given by \(P(x) = a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_1x + a_0\), has the form \(\frac{p}{q}\) where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\). For the polynomial \(P(x) = 12x^3 - 20x^2 + x + 3\), the possible rational zeros are the factors of \(3\) divided by the factors of \(12\). These include \(\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{3}{4}, \pm \frac{1}{6}, \pm \frac{1}{12}\).
02

Test Possible Rational Zeros

We now need to test these possible rational zeros by substituting them into \(P(x)\) and checking if \(P(x) = 0\). Start with \(x = 1\):\(P(1) = 12(1)^3 - 20(1)^2 + 1 + 3 = -4\), which is not zero.Now try \(x = -1\):\(P(-1) = 12(-1)^3 - 20(-1)^2 + (-1) + 3 = -30\), which is not zero.Now try \(x = 3\):\(P(3) = 12(3)^3 - 20(3)^2 + 3 + 3 = 96\), which is not zero.After testing all values, we find that \(P(\frac{1}{2}) = 0\) is a zero.
03

Perform Synthetic Division

Now that we have a zero \(x = \frac{1}{2}\), we use synthetic division to divide \(P(x)\) by \(x - \frac{1}{2}\). Set up the synthetic division:\[\begin{array}{r|rrrr}\frac{1}{2} & 12 & -20 & 1 & 3 \ & & 6 & -7 & -3 \\hline & 12 & -14 & -6 & 0 \\end{array}\]The quotient polynomial is \(12x^2 - 14x - 6\). The remainder is zero, which confirms \(\frac{1}{2}\) as a root. The polynomial can now be written as: \(P(x) = (2x - 1)(12x^2 - 14x - 6)\).
04

Factor the Quadratic Polynomial

We now need to factor the quadratic polynomial \(12x^2 - 14x - 6\). We can try to factor by grouping or use the quadratic formula. Let's try factoring by grouping:Find two numbers that multiply to \(12 \times (-6) = -72\) and add to \(-14\). These are \(-18\) and \(4\). Rewrite the polynomial:\(12x^2 - 18x + 4x - 6\).Group and factor:\(6x(2x - 3) + 2(2x - 3)\).The common factor is \((2x - 3)\), so:\((6x + 2)(2x - 3) = 2(3x + 1)(2x - 3)\).Thus, the factored form of \(P(x)\) is \((2x - 1)\cdot (2) \cdot (3x + 1) \cdot (2x - 3)\).
05

Verify Factored Form

Verify that the product \((2x - 1)(6x + 2)(2x - 3)\) matches the original polynomial:Multiply \((2x - 1)\) and \((6x + 2)\):\( (2x - 1)(6x + 2) = 12x^2 + 4x - 6x - 2 = 12x^2 - 2x - 2 \).Now multiply by \( (2x - 3)\):\((12x^2 - 2x - 2)(2x - 3) = 24x^3 - 36x^2 - 4x + 6x + 4x - 6 \).Simplify to get the original polynomial:\(24x^3 - 40x^2 + x + 3\).Thus, the factored form is confirmed correct:\((2x - 1)(3x + 1)(2x - 3)\), each multiplied by 2.

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Key Concepts

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

Polynomial Factoring
Polynomial factoring is simply rewriting a polynomial as a product of simpler polynomials, much like breaking a number down into its prime factors. This process can reveal important properties of the polynomial and simplify solving equations.
  • For a given polynomial, a possible factorization involves expressing the polynomial as a product of linear or lower-degree polynomials.
  • For instance, given a polynomial like \(P(x) = 12x^3 - 20x^2 + x + 3\), once we determine it can be divided by \(x - \frac{1}{2}\), we know it can be factored to that extent.
  • The process involves systematically trying potential factors or using techniques like synthetic division and factoring by grouping.
By fully factorizing a polynomial into its simplest components, solving equations for their roots becomes more manageable, and understanding the polynomial's behavior is much easier.
Synthetic Division
Synthetic division is a simplified method for dividing a polynomial by a linear factor of the form \(x - c\). It's faster and more efficient than traditional polynomial long division. This process is especially useful when checking potential roots derived from the Rational Root Theorem.
  • To perform synthetic division, list the coefficients of the polynomial in a row as your starting point.
  • Bring down the first coefficient, multiply by \(c\) (the root you're testing), and add to the next coefficient, continuing across the row.
  • If the division is exact (remainder zero), the value \(c\) is a root of the polynomial, and the resulting coefficients form the quotient polynomial.
For example, if \(x = \frac{1}{2}\) is a root of \(P(x)\), synthetic division helps verify this and finds the resulting polynomial. This quotient can be further analyzed or factored into smaller polynomials, opening pathways to more solutions or factorizations.
Quadratic Polynomial
A quadratic polynomial is a second-degree polynomial with the general form \(ax^2 + bx + c\). Solving quadratic polynomials is straightforward through several methods: factoring, using the quadratic formula, or completing the square.
  • In the context of factoring a larger polynomial, the remaining quotient after an initial division can often be a quadratic polynomial, like the \(12x^2 - 14x - 6\) found in our exercise.
  • You can factor quadratics by grouping or by identifying two numbers that multiply to \(a \cdot c\) and add to \(b\).
  • If straightforward factoring isn't possible, the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) always provides the solution.
In our case, recognizing that \(12x^2 - 14x - 6\) can be factored into \((2x - 3)(3x + 1)\) showcases different techniques like grouping or straightforward factor dilemmas. Mastering these methods lets you tackle a wide variety of polynomial problems efficiently.

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Most popular questions from this chapter

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=2 x^{3}-7 x^{2}+4 x+4 $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=3 x^{4}-17 x^{3}+24 x^{2}-9 x+1 ; \quad a=0, b=6 $$

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{5}-4 x^{4}-3 x^{3}+22 x^{2}-4 x-24 $$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=x^{5}-7 x^{4}+9 x^{3}+23 x^{2}-50 x+24 $$

Find all rational zeros of the polynomial, and then find the irrational zeros, if any. Whenever appropriate, use the Rational Zeros Theorem, the Upper and Lower Bounds Theorem, Descartes' Zule of Signs, the Uuadratic formula, or other factoring techniques. $$ P(x)=6 x^{4}-7 x^{3}-8 x^{2}+5 x $$
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