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Chapter 3: Problem 36

You are given a polynomial and one of its zeros. Use the techniques in this section to find the rest of the real zeros and factor the polynomial. \(2 x^{3}-x^{2}-10 x+5, \quad c=\frac{1}{2}\)

Short Answer

Expert verified
The polynomial is factored as \((x - \frac{1}{2}) \times 2(x - \sqrt{5})(x + \sqrt{5})\).

Step by step solution

01

Verify Given Zero

Check that \(c = \frac{1}{2}\) is a zero of the polynomial \(2x^3 - x^2 - 10x + 5\). Substitute \(x = \frac{1}{2}\) into the polynomial: \[2\left(\frac{1}{2}\right)^3 - \left(\frac{1}{2}\right)^2 - 10\left(\frac{1}{2}\right) + 5 = 0.\]Calculating this gives \(2 \times \frac{1}{8} - \frac{1}{4} - 5 + 5 = 0\), confirming that \(\frac{1}{2}\) is a zero.
02

Perform Synthetic Division

Divide the polynomial \(2x^3 - x^2 - 10x + 5\) by \(x - \frac{1}{2}\) using synthetic division:- Bring down the leading coefficient \(2\).- Multiply by \(\frac{1}{2}\) and add to the next coefficient \(-1\), resulting in \(0\).- Repeat with \(0\) to \(-10\) giving \(-10\).- Lastly, multiply and add to \(5\), which gives 0, confirming division without remainder.The quotient is \(2x^2 - 10\).
03

Factor the Quotient Polynomial

The polynomial remaining after division is \(2x^2 - 10\). Factor this:\[2(x^2 - 5).\]The expression \(x^2 - 5\) can be written as the difference of squares, \((x - \sqrt{5})(x + \sqrt{5})\). So, \(2(x - \sqrt{5})(x + \sqrt{5})\) is the factored form.
04

Write the Completely Factored Form

The complete factorization of the original polynomial is:\[(x - \frac{1}{2}) \times 2(x - \sqrt{5})(x + \sqrt{5}).\]Thus, it is factored into linear factors.

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Key Concepts

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

Understanding Synthetic Division
Synthetic division is like a shortcut for dividing a polynomial by a binomial of the form \(x - c\), where \(c\) is a constant. Standard polynomial long division can be lengthy, but synthetic division simplifies this process. You only need the coefficients of the polynomial and the zero you are checking.Here's how you perform synthetic division:
  • Write down the constant zero of the form \(x - c\), placing \(c\) outside the division bracket.
  • List the coefficients of the polynomial inside the division bracket. For our example, the coefficients are \(2, -1, -10, 5\).
  • Bring down the first coefficient following the division bracket, which is 2.
  • Multiply this leading coefficient by \(\frac{1}{2}\) (our \(c\)) and add the result to the next coefficient.
  • Continue this process for all coefficients.
If you end with zero as the remainder, it confirms that \(c\) is indeed a zero of the polynomial. This method provides the quotient easily which we further use to factor and find other zeros.
Mastering Polynomial Factorization
Factorization is the process of breaking down a complex expression into simpler components or factors. For polynomials, we often aim to express them as the product of lower-degree polynomials. This process simplifies solving polynomial equations and finding zeros.In our exercise, after using synthetic division, we ended up with the quadratic polynomial \(2x^2 - 10\). Notice that this expression can be further simplified by factoring out the common factor of 2:\[2(x^2 - 5)\]From here, we need to see if the quadratic \(x^2 - 5\) can be broken down further. This is where difference of squares comes in, allowing even further factorization.
Exploring the Difference of Squares
The difference of squares is a special factorization technique. It is based on the algebraic identity: \(a^2 - b^2 = (a - b)(a + b)\). This identity helps simplify expressions where the polynomial is expressible as one square minus another.Applying this to our polynomial factor \(x^2 - 5\), we recognize:
  • Here, \(a^2 = x^2\) and \(b^2 = 5\) or \(b = \sqrt{5}\).
  • Rearrange \(x^2 - 5\), factoring it as \((x - \sqrt{5})(x + \sqrt{5})\).
By recognizing and employing the difference of squares technique, students can factor seemingly complex polynomials, further simplifying finding zeros. This approach helps break down quadratic factors into products of linear factors.

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Most popular questions from this chapter

For the given polynomial: \- Use Cauchy's Bound to find an interval containing all of the real zeros. \- Use the Rational Zeros Theorem to make a list of possible rational zeros. \- Use Descartes' Rule of Signs to list the possible number of positive and negative real zeros, counting multiplicities. \(f(x)=x^{3}+4 x^{2}-11 x+6\)

According to US Postal regulations, a rectangular shipping box must satisfy the inequality "Length \(+\) Girth \(\leq 130\) inches" for Parcel Post and "Length \(+\) Girth \(\leq 108\) inches" for other services. Let's assume we have a closed rectangular box with a square face of side length \(x\) as drawn below. The length is the longest side and is clearly labeled. The girth is the distance around the box in the other two dimensions so in our case it is the sum of the four sides of the square, \(4 x\). (a) Assuming that we'll be mailing a box via Parcel Post where Length \(+\) Girth \(=130\) inches, express the length of the box in terms of \(x\) and then express the volume \(V\) of the box in terms of \(x\). (b) Find the dimensions of the box of maximum volume that can be shipped via Parcel Post. (c) Repeat parts \(33 \mathrm{a}\) and \(33 \mathrm{~b}\) if the box is shipped using "other services".

Use synthetic division to perform the indicated division. Write the polynomial in the form \(p(x)=d(x) q(x)+r(x)\). \(\left(x^{3}+8\right) \div(x+2)\)

Solve the polynomial inequality and state your answer using interval notation. \(3 x^{2}+2 x

Find the real zeros of the polynomial using the techniques specified by your instructor. State the multiplicity of each real zero. \(f(x)=x^{4}+2 x^{2}-15\)
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