Splash Screen

Five-Minute Check (over Lesson 5–7) CCSS Then/Now Key Concept: Rational Zero Theorem Example 1: Identify Possible Zeros Example 2: Real-World Example: Find Rational Zeros Example 3: Find All Zeros Lesson Menu

Solve x2 + 4x + 7 = 0. A. B. C. D. 5-Minute Check 1

Solve x2 + 4x + 7 = 0. A. B. C. D. 5-Minute Check 1

What best describes the roots of the equation 2x3 + 5x2 – 23x + 10 = 0? A. 3 imaginary B. 2 imaginary C. 3 real D. 2 real 5-Minute Check 2

What best describes the roots of the equation 2x3 + 5x2 – 23x + 10 = 0? A. 3 imaginary B. 2 imaginary C. 3 real D. 2 real 5-Minute Check 2

How many negative real zeros does p(x) = x4 – 7x3 + 2x2 – 6x – 2 have? B. 2 C. 1 D. 0 5-Minute Check 3

How many negative real zeros does p(x) = x4 – 7x3 + 2x2 – 6x – 2 have? B. 2 C. 1 D. 0 5-Minute Check 3

What is the least degree of a polynomial function with zeros that include 5 and 3i? B. 3 C. 2 D. 1 5-Minute Check 4

What is the least degree of a polynomial function with zeros that include 5 and 3i? B. 3 C. 2 D. 1 5-Minute Check 4

Which of the following is not a zero of 4x3 + 9x2 + 22x + 5? B. C. –1 + 2i D. –1 – 2i 5-Minute Check 5

Which of the following is not a zero of 4x3 + 9x2 + 22x + 5? B. C. –1 + 2i D. –1 – 2i 5-Minute Check 5

Mathematical Practices 8 Look for and express regularity in repeated reasoning. CCSS

Identify possible rational zeros of a polynomial function. You found zeros of quadratic functions of the form f(x) = ax 2 + bx + c. Identify possible rational zeros of a polynomial function. Find all of the rational zeros of a polynomial function. Then/Now

Concept

A. List all of the possible rational zeros of f(x) = 3x4 – x3 + 4. Identify Possible Zeros A. List all of the possible rational zeros of f(x) = 3x4 – x3 + 4. Answer: Example 1

A. List all of the possible rational zeros of f(x) = 3x4 – x3 + 4. Identify Possible Zeros A. List all of the possible rational zeros of f(x) = 3x4 – x3 + 4. Answer: Example 1

B. List all of the possible rational zeros of f(x) = x4 + 7x3 – 15. Identify Possible Zeros B. List all of the possible rational zeros of f(x) = x4 + 7x3 – 15. Since the coefficient of x4 is 1, the possible zeros must be a factor of the constant term –15. Answer: Example 1

B. List all of the possible rational zeros of f(x) = x4 + 7x3 – 15. Identify Possible Zeros B. List all of the possible rational zeros of f(x) = x4 + 7x3 – 15. Since the coefficient of x4 is 1, the possible zeros must be a factor of the constant term –15. Answer: So, the possible rational zeros are ±1, ±3, ±5, and ±15. Example 1

A. List all of the possible rational zeros of f(x) = 2x3 + x + 6. C. D. Example 1

A. List all of the possible rational zeros of f(x) = 2x3 + x + 6. C. D. Example 1

B. List all of the possible rational zeros of f(x) = x3 + 3x + 24. C. D. Example 1

B. List all of the possible rational zeros of f(x) = x3 + 3x + 24. C. D. Example 1

Let x = the height, x – 2 = the width, and x + 4 = the length. Find Rational Zeros GEOMETRY The volume of a rectangular solid is 1120 cubic feet. The width is 2 feet less than the height, and the length is 4 feet more than the height. Find the dimensions of the solid. Let x = the height, x – 2 = the width, and x + 4 = the length. Example 2

Write the equation for volume. Find Rational Zeros Write the equation for volume. ℓ ● w ● h = V Formula for volume Substitute. Multiply. Subtract 1120 from each side. The leading coefficient is 1, so the possible integer zeros are factors of 1120. Since length can only be positive, we only need to check positive zeros. Example 2

Find Rational Zeros The possible factors are 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 32, 35, 40, 56, 70, 80, 112, 140, 160, 224, 280, 560, and 1120. By Descartes’ Rule of Signs, we know that there is exactly one positive real root. Make a table and test possible real zeros. So, the zero is 10. The other dimensions are 10 – 2 or 8 feet and 10 + 4 or 14 feet. Example 2

Find Rational Zeros Answer: Example 2

Answer: ℓ = 14 ft, w = 8 ft, and h = 10 ft Find Rational Zeros Answer: ℓ = 14 ft, w = 8 ft, and h = 10 ft Check Verify that the dimensions are correct. 10 × 8 × 14 = 1120  Example 2

GEOMETRY The volume of a rectangular solid is 100 cubic feet GEOMETRY The volume of a rectangular solid is 100 cubic feet. The width is 3 feet less than the height and the length is 5 feet more than the height. What are the dimensions of the solid? A. h = 6, ℓ = 11, w = 3 B. h = 5, ℓ = 10, w = 2 C. h = 7, ℓ = 12, w = 4 D. h = 8, ℓ = 13, w = 5 Example 2

GEOMETRY The volume of a rectangular solid is 100 cubic feet GEOMETRY The volume of a rectangular solid is 100 cubic feet. The width is 3 feet less than the height and the length is 5 feet more than the height. What are the dimensions of the solid? A. h = 6, ℓ = 11, w = 3 B. h = 5, ℓ = 10, w = 2 C. h = 7, ℓ = 12, w = 4 D. h = 8, ℓ = 13, w = 5 Example 2

Find all of the zeros of f(x) = x4 + x3 – 19x2 + 11x + 30. Find All Zeros Find all of the zeros of f(x) = x4 + x3 – 19x2 + 11x + 30. From the corollary to the Fundamental Theorem of Algebra, we know there are exactly 4 complex roots. According to Descartes’ Rule of Signs, there are 2 or 0 positive real roots and 2 or 0 negative real roots. The possible rational zeros are 1, 2, 3, 5, 6, 10, 15, and 30. Make a table and test some possible rational zeros. Example 3

Find All Zeros Since f(2) = 0, you know that x = 2 is a zero. The depressed polynomial is x3 + 3x2 – 13x – 15. Example 3

Find All Zeros Since x = 2 is a positive real zero, and there can only be 2 or 0 positive real zeros, there must be one more positive real zero. Test the next possible rational zeros on the depressed polynomial. There is another zero at x = 3. The depressed polynomial is x2 + 6x + 5. Example 3

Write the depressed polynomial. Find All Zeros Factor x2 + 6x + 5. Write the depressed polynomial. Factor. Zero Product Property or There are two more real roots at x = –5 and x = –1. Answer: Example 3

Write the depressed polynomial. Find All Zeros Factor x2 + 6x + 5. Write the depressed polynomial. Factor. Zero Product Property or There are two more real roots at x = –5 and x = –1. Answer: The zeros of this function are –5, –1, 2, and 3. Example 3

Find all of the zeros of f(x) = x4 + 4x3 – 14x2 – 36x + 45. A. –10, –3, 1, and 3 B. –5, 1, and 3 C. –5 and –3 D. –5, –3, 1 and 3 Example 3

Find all of the zeros of f(x) = x4 + 4x3 – 14x2 – 36x + 45. A. –10, –3, 1, and 3 B. –5, 1, and 3 C. –5 and –3 D. –5, –3, 1 and 3 Example 3

End of the Lesson