Math Problem Statement

1. Find the remainder when 4x^3 - 5x + 1 is divided by: a) x - 2 b) x + 3 c) 2x - 1 2. The expression 4x^2 - px + 7 leaves a remainder of -2 when divided by x - 3. Find the value of p.

Solution

Let's go through each question one at a time.

Question 1

To find the remainder when 4x35x+14x^3 - 5x + 1 is divided by each of the expressions, we can use the Remainder Theorem. According to this theorem, if we want to find the remainder of a polynomial f(x)f(x) when divided by xax - a, we evaluate f(a)f(a).

Given f(x)=4x35x+1f(x) = 4x^3 - 5x + 1:

a) x2x - 2

To find the remainder when f(x)f(x) is divided by x2x - 2, we substitute x=2x = 2: f(2)=4(2)35(2)+1=4810+1=3210+1=23f(2) = 4(2)^3 - 5(2) + 1 = 4 \cdot 8 - 10 + 1 = 32 - 10 + 1 = 23 So, the remainder is 23.

b) x+3x + 3

To find the remainder when f(x)f(x) is divided by x+3x + 3, we substitute x=3x = -3: f(3)=4(3)35(3)+1=4(27)+15+1=108+15+1=92f(-3) = 4(-3)^3 - 5(-3) + 1 = 4 \cdot (-27) + 15 + 1 = -108 + 15 + 1 = -92 The remainder is -92.

c) 2x12x - 1

When dividing by 2x12x - 1, we substitute x=12x = \frac{1}{2}: f(12)=4(12)35(12)+1=41852+1=1252+1=1f\left(\frac{1}{2}\right) = 4\left(\frac{1}{2}\right)^3 - 5\left(\frac{1}{2}\right) + 1 = 4 \cdot \frac{1}{8} - \frac{5}{2} + 1 = \frac{1}{2} - \frac{5}{2} + 1 = -1 The remainder is -1.

Question 2

We need to determine the value of pp such that the expression 4x2px+74x^2 - px + 7 leaves a remainder of 2-2 when divided by x3x - 3.

Using the Remainder Theorem, we substitute x=3x = 3 and set the result equal to 2-2: f(3)=4(3)2p(3)+7=2f(3) = 4(3)^2 - p(3) + 7 = -2 493p+7=24 \cdot 9 - 3p + 7 = -2 363p+7=236 - 3p + 7 = -2 433p=243 - 3p = -2 3p=45-3p = -45 p=15p = 15 So, the value of pp is 15.

Would you like more details or have any further questions?


Related Questions:

  1. How does the Remainder Theorem apply to polynomials of higher degrees?
  2. What is the difference between the Remainder Theorem and Synthetic Division?
  3. How would we find the quotient in addition to the remainder?
  4. What changes when dividing by a binomial like 2x12x - 1 instead of xax - a?
  5. How would we determine the remainder if the divisor was x24x^2 - 4?

Tip: When using the Remainder Theorem, always check if the divisor is of the form xax - a for straightforward substitution.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Remainder Theorem

Formulas

Remainder Theorem: f(a) when dividing by (x - a)

Theorems

Remainder Theorem

Suitable Grade Level

Grade 10-12