The functions x4 + bx 3×2 48 5 and 3×3 8×2 2x 13, have the same remainder

when divided by (x – 2). What is the value of b?

The functions x3 7x – 4 and 3×3 3×2 + bx + 14, have the same remainder when divided

by (x-3). What is the value of b?

The function f(x) = ax2 + x – 7 has a remainder of 3 when divided by (x – 2).

Find the value of a.

Objectives:

– To solve problems involving polynomial remainders.

– To determine the value of a variable within a polynomial equation.

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Learning Outcomes:

By the end of this content, the learner will be able to:

– Solve polynomial remainder problems involving linear factors.

– Compute the value of a variable in polynomial equations using the remainder theorem.

Heading 1: Polynomial Remainders

– Understand the concept of polynomial remainders.

– Identify the linear factor that divides two polynomial functions that have identical remainders.

Heading 2: Solving for a Variable in Polynomial Equations

– Utilize the remainder theorem to solve for a variable in a polynomial equation.

– Solve problems related to the computation for the value of a variable in quadratic equations with remainders.

Solution 1:

To find the value of b, we need to first determine the remainder when both functions x4 + bx3x2 + 48x + 5 and 3×3 + 8×2 + 2x + 13 are divided by (x – 2). Let R(x) be the remainder function. Then we have:

x4 + bx3x2 + 48x + 5 = (x – 2)q(x) + R(x)

3×3 + 8×2 + 2x + 13 = (x – 2)p(x) + R(x)

where q(x) and p(x) are quotient functions. Since both functions have the same remainder, we can subtract the two equations above to obtain:

x4 + bx3x2 + 48x + 5 – (3×3 + 8×2 + 2x + 13) = (x – 2)(q(x) – p(x))

Simplifying the left-hand side gives:

x4 + bx3x2 – 3×3 + 40x – 8 = (x – 2)(q(x) – p(x))

At x = 2, the left-hand side becomes:

16 + 4b – 6 + 80 – 8 = 4b + 82

At x = 2, the right-hand side becomes:

0

Since the two functions have the same remainder, the right-hand side must be 0. Therefore, we have:

4b + 82 = 0

4b = -82

b = -20.5

Hence, Solution 1: The value of b is -20.5.

Solution 2:

To find the value of b, we need to first determine the remainder when both functions x3 + 7x – 4 and 3×3 + 3×2 + bx + 14 are divided by (x – 3). Let R(x) be the remainder function. Then we have:

x3 + 7x – 4 = (x – 3)q(x) + R(x)

3×3 + 3×2 + bx + 14 = (x – 3)p(x) + R(x)

where q(x) and p(x) are quotient functions. Since both functions have the same remainder, we can subtract the two equations above to obtain:

x3 + 7x – 4 – (3×3 + 3×2 + bx + 14) = (x – 3)(q(x) – p(x))

Simplifying the left-hand side gives:

-2×3 + 4×2 – bx – 18 = (x – 3)(q(x) – p(x))

At x = 3, the left-hand side becomes:

-18 – 3b

At x = 3, the right-hand side becomes:

0

Since the two functions have the same remainder, the right-hand side must be 0. Therefore, we have:

-18 – 3b = 0

-3b = 18

b = -6

Hence, Solution 2: The value of b is -6.

Solution 3:

We are given that the function f(x) = ax2 + x – 7 has a remainder of 3 when divided by (x – 2). Let R(x) be the remainder function. Then we have:

ax2 + x – 7 = (x – 2)q(x) + R(x)

where q(x) is the quotient function. At x = 2, the left-hand side becomes:

4a + 2 – 7 = 4a – 5

At x = 2, the right-hand side becomes:

R(2) = 3

Since the remainder is 3, we have:

R(2) = -2a + 2 – 7 = 3

-2a – 5 = 3

-2a = 8

a = -4

Hence, Solution: The value of a is -4.

Suggested Resources/Books:

– “Polynomial Remainder Theorem” by Khan Academy

– “Algebra and Trigonometry” by Robert F. Blitzer

– “College Algebra” by Raymond A. Barnett

Similar Asked Questions:

1. How to find the remainder of a polynomial when divided by a given linear factor?

2. What is the significance of the remainder in polynomial division?

3. Can the remainder of a polynomial be greater than the divisor?

4. How to find the value of a variable in a polynomial given the remainder and divisor?

5. What is the relationship between remainders and factors in polynomial division?The functions x4 + bx 3×2 48 5 and 3×3 8×2 2x 13, have the same remainder

when divided by (x – 2). What is the value of b?

The functions x3 7x – 4 and 3×3 3×2 + bx + 14, have the same remainder when divided

by (x-3). What is the value of b?

The function f(x) = ax2 + x – 7 has a remainder of 3 when divided by (x – 2).

Find the value of a.

Objectives:

– To solve problems involving polynomial remainders.

– To determine the value of a variable within a polynomial equation.

Learning Outcomes:

By the end of this content, the learner will be able to:

– Solve polynomial remainder problems involving linear factors.

– Compute the value of a variable in polynomial equations using the remainder theorem.

Heading 1: Polynomial Remainders

– Understand the concept of polynomial remainders.

– Identify the linear factor that divides two polynomial functions that have identical remainders.

Heading 2: Solving for a Variable in Polynomial Equations

– Utilize the remainder theorem to solve for a variable in a polynomial equation.

– Solve problems related to the computation for the value of a variable in quadratic equations with remainders.

Solution 1:

To find the value of b, we need to first determine the remainder when both functions x4 + bx3x2 + 48x + 5 and 3×3 + 8×2 + 2x + 13 are divided by (x – 2). Let R(x) be the remainder function. Then we have:

x4 + bx3x2 + 48x + 5 = (x – 2)q(x) + R(x)

3×3 + 8×2 + 2x + 13 = (x – 2)p(x) + R(x)

where q(x) and p(x) are quotient functions. Since both functions have the same remainder, we can subtract the two equations above to obtain:

x4 + bx3x2 + 48x + 5 – (3×3 + 8×2 + 2x + 13) = (x – 2)(q(x) – p(x))

Simplifying the left-hand side gives:

x4 + bx3x2 – 3×3 + 40x – 8 = (x – 2)(q(x) – p(x))

At x = 2, the left-hand side becomes:

16 + 4b – 6 + 80 – 8 = 4b + 82

At x = 2, the right-hand side becomes:

0

Since the two functions have the same remainder, the right-hand side must be 0. Therefore, we have:

4b + 82 = 0

4b = -82

b = -20.5

Hence, Solution 1: The value of b is -20.5.

Solution 2:

To find the value of b, we need to first determine the remainder when both functions x3 + 7x – 4 and 3×3 + 3×2 + bx + 14 are divided by (x – 3). Let R(x) be the remainder function. Then we have:

x3 + 7x – 4 = (x – 3)q(x) + R(x)

3×3 + 3×2 + bx + 14 = (x – 3)p(x) + R(x)

x3 + 7x – 4 – (3×3 + 3×2 + bx + 14) = (x – 3)(q(x) – p(x))

Simplifying the left-hand side gives:

-2×3 + 4×2 – bx – 18 = (x – 3)(q(x) – p(x))

At x = 3, the left-hand side becomes:

-18 – 3b

At x = 3, the right-hand side becomes:

0

Since the two functions have the same remainder, the right-hand side must be 0. Therefore, we have:

-18 – 3b = 0

-3b = 18

b = -6

Hence, Solution 2: The value of b is -6.

Solution 3:

We are given that the function f(x) = ax2 + x – 7 has a remainder of 3 when divided by (x – 2). Let R(x) be the remainder function. Then we have:

ax2 + x – 7 = (x – 2)q(x) + R(x)

where q(x) is the quotient function. At x = 2, the left-hand side becomes:

4a + 2 – 7 = 4a – 5

At x = 2, the right-hand side becomes:

R(2) = 3

Since the remainder is 3, we have:

R(2) = -2a + 2 – 7 = 3

-2a – 5 = 3

-2a = 8

a = -4

Hence, Solution: The value of a is -4.

Suggested Resources/Books:

– “Polynomial Remainder Theorem” by Khan Academy

– “Algebra and Trigonometry” by Robert F. Blitzer

– “College Algebra” by Raymond A. Barnett

Similar Asked Questions:

1. How to find the remainder of a polynomial when divided by a given linear factor?

2. What is the significance of the remainder in polynomial division?

3. Can the remainder of a polynomial be greater than the divisor?

4. How to find the value of a variable in a polynomial given the remainder and divisor?

5. What is the relationship between remainders and factors in polynomial division?

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