What is the value of a if the function has a remainder of 3 when divided by (x – ?

  

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.

Don't use plagiarized sources. Get Your Custom Essay on
What is the value of a if the function has a remainder of 3 when divided by (x – ?
Just from $13/Page
Order Essay

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)

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?

Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more
× How can I help you?