Problem 25
Question
Find all of the zeros of each function. \(p(x)=x^{3}+3 x^{2}-25 x+21\)
Step-by-Step Solution
Verified Answer
The zeros are \(x = 1\), \(x = 3\), and \(x = -7\).
1Step 1: Identify Possible Rational Zeros
According to the Rational Root Theorem, the possible rational zeros of \(p(x)\) are the factors of the constant term \(21\) (i.e., \(\pm 1, \pm 3, \pm 7, \pm 21\)) divided by the factors of the leading coefficient \(1\) (i.e., \(\pm 1\)). Thus, the potential rational zeros are \(\pm 1, \pm 3, \pm 7, \pm 21\).
2Step 2: Test Potential Zeros Using Synthetic Division
We use synthetic division to test the possible zeros. Testing \(x = 1\) first: - The coefficients of the polynomial are \(1, 3, -25, 21\). - Perform synthetic division: 1 \( | \) 1, 3, -25, 21, resulting in a remainder of 0. Thus, \(x = 1\) is a zero of \(p(x)\) and \((x - 1)\) is a factor.
3Step 3: Factor Polynomial Using Found Zero
Since \(x = 1\) is a zero, we divide \(p(x)\) by \((x - 1)\) using synthetic division:Performing the division:1 \( | \) 1, 3, -25, 21 \( \) \( \text{1, 4, -21, 0}\)The result is \(x^2 + 4x - 21\). So, \(p(x) = (x - 1)(x^2 + 4x - 21)\).
4Step 4: Factor the Quadratic
Factor the quadratic \(x^2 + 4x - 21\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 4\), and \(c = -21\):\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-21)}}{2 \cdot 1} \]\[ x = \frac{-4 \pm \sqrt{16 + 84}}{2} \]\[ x = \frac{-4 \pm \sqrt{100}}{2} \]\[ x = \frac{-4 \pm 10}{2} \]The solutions are \(x = 3\) and \(x = -7\). Thus, \(x^2 + 4x - 21 = (x - 3)(x + 7)\).
5Step 5: List All Zeros
Now, we rewrite the full factorization of \(p(x)\) as \((x - 1)(x - 3)(x + 7)\). The zeros of \(p(x)\) are \(x = 1\), \(x = 3\), and \(x = -7\).
Key Concepts
Rational Root TheoremSynthetic DivisionQuadratic FormulaFactoring Polynomials
Rational Root Theorem
The Rational Root Theorem is a valuable tool for finding potential rational zeros of a polynomial. It states that any rational solution, or zero, of a polynomial equation with integer coefficients will be a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
In our problem, the polynomial is \( p(x) = x^3 + 3x^2 - 25x + 21 \). The constant term is \( 21 \), and the leading coefficient is \( 1 \). This makes the potential rational zeros the factors of \( 21 \), which are:
In our problem, the polynomial is \( p(x) = x^3 + 3x^2 - 25x + 21 \). The constant term is \( 21 \), and the leading coefficient is \( 1 \). This makes the potential rational zeros the factors of \( 21 \), which are:
- \( \pm 1 \)
- \( \pm 3 \)
- \( \pm 7 \)
- \( \pm 21 \)
Synthetic Division
Synthetic division is a simplified form of polynomial division, particularly useful for dividing by linear factors. It quickly determines if a given number is a zero of the polynomial.
Here's how it's utilized in the exercise: first, we choose a potential zero, say \( x = 1 \). Then, we lay out the coefficients of the polynomial \(1, 3, -25, 21\) for synthetic division. The division process checks if there is a zero remainder after using these coefficients. Since the remainder is zero for \( x = 1 \), it confirms that \( x = 1 \) is a zero and \( (x - 1) \) is a factor of the polynomial.
This is a crucial step before proceeding to factor or apply other formulas.
Here's how it's utilized in the exercise: first, we choose a potential zero, say \( x = 1 \). Then, we lay out the coefficients of the polynomial \(1, 3, -25, 21\) for synthetic division. The division process checks if there is a zero remainder after using these coefficients. Since the remainder is zero for \( x = 1 \), it confirms that \( x = 1 \) is a zero and \( (x - 1) \) is a factor of the polynomial.
This is a crucial step before proceeding to factor or apply other formulas.
Quadratic Formula
The Quadratic Formula is a reliable method for finding the roots of any quadratic equation \( ax^2 + bx + c \). Given by the formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In our exercise, after identifying \( x = 1 \) as a zero and factoring it out, the remaining polynomial was \( x^2 + 4x - 21 \). For this quadratic, substituting \( a = 1 \), \( b = 4 \), and \( c = -21 \) into the formula provides solutions \( x = 3 \) and \( x = -7 \).
This step allows us to complete the factoring process and find the remaining zeros efficiently, confirming our polynomial factorization.
In our exercise, after identifying \( x = 1 \) as a zero and factoring it out, the remaining polynomial was \( x^2 + 4x - 21 \). For this quadratic, substituting \( a = 1 \), \( b = 4 \), and \( c = -21 \) into the formula provides solutions \( x = 3 \) and \( x = -7 \).
This step allows us to complete the factoring process and find the remaining zeros efficiently, confirming our polynomial factorization.
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of its simplest factors. It is an essential technique for simplification and solving polynomial equations.
In the given exercise, factoring begins once we find a zero using synthetic division. The polynomial \( p(x) \) initially factors as \((x - 1)(x^2 + 4x - 21)\) after confirming \( x = 1 \) is a zero.
Next, using the quadratic formula on \( x^2 + 4x - 21 \), we factor further to \( (x - 3)(x + 7) \). Putting it all together, the complete factorization of \( p(x) \) is \((x - 1)(x - 3)(x + 7)\).
In the given exercise, factoring begins once we find a zero using synthetic division. The polynomial \( p(x) \) initially factors as \((x - 1)(x^2 + 4x - 21)\) after confirming \( x = 1 \) is a zero.
Next, using the quadratic formula on \( x^2 + 4x - 21 \), we factor further to \( (x - 3)(x + 7) \). Putting it all together, the complete factorization of \( p(x) \) is \((x - 1)(x - 3)(x + 7)\).
- This shows that the zeros of the polynomial are \( x = 1 \), \( x = 3 \), and \( x = -7 \).
Other exercises in this chapter
Problem 24
Simplify. Assume that no variable equals 0. $$ \left(-2 r^{2} s\right)^{3}\left(3 r s^{2}\right) $$
View solution Problem 25
Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 2 x^{3}+17 x^{2}+23 x-42 ; 2 x+7
View solution Problem 25
Find all of the zeros of each function. \(f(x)=x^{3}-5 x^{2}-7 x+51\)
View solution Problem 25
Write each expression in quadratic form, if possible. $$ a^{8}+10 a^{2}-16 $$
View solution