Step-by-step explanation:

To find the value of k, we need to divide the given polynomials p(x) and f(x) by (x-4) and then equate the remainders to find a relation between k, R₁, and R₂. Since both p(x) and f(x) are being divided by (x-4), the remainders will be the values of p(4) and f(4) respectively.

Let's start by finding the remainders R₁ and R₂:

Remainder R₁ for p(x) = kx² + 3x² - 13 when divided by (x-4):

To find R₁, evaluate p(x) at x = 4:

R₁ = p(4) = k(4)² + 3(4)² - 13

R₁ = 16k + 48 - 13

R₁ = 16k + 35

Remainder R₂ for f(x) = 2x³ - 5x - k when divided by (x-4):

To find R₂, evaluate f(x) at x = 4:

R₂ = f(4) = 2(4)³ - 5(4) - k

R₂ = 2(64) - 20 - k

R₂ = 128 - 20 - k

R₂ = 108 - k

Now, according to the given information, the sum of R₁ and R₂ is equal to 269:

R₁ + R₂ = 269

(16k + 35) + (108 - k) = 269

Now, solve for k:

16k + 35 + 108 - k = 269

15k + 143 = 269

Subtract 143 from both sides:

15k = 126

Now, divide both sides by 15 to solve for k:

k = 126 / 15

k = 8.4

So, the value of k is 8.4

Answer:

To find the remainders R₁ and R₂ when dividing the polynomials p(x) and f(x) by (x-4), we can use the Remainder Theorem, which states that when a polynomial f(x) is divided by (x-a), the remainder is f(a).

1. For p(x) = kx² + 3x² - 13:

When dividing p(x) by (x-4), we substitute x = 4 into the polynomial:

R₁ = p(4) = k(4)² + 3(4)² - 13 = 16k + 48 - 13 = 16k + 35.

2. For f(x) = 2x³ - 5x - k:

When dividing f(x) by (x-4), we substitute x = 4 into the polynomial:

R₂ = f(4) = 2(4)³ - 5(4) - k = 2(64) - 20 - k = 128 - 20 - k = 108 - k.

Now, we are given that R₁ + R₂ = 269. Let's substitute the expressions for R₁ and R₂:

(16k + 35) + (108 - k) = 269

Now, let's solve for k:

16k + 35 + 108 - k = 269

Combine like terms:

15k + 143 = 269

Subtract 143 from both sides:

15k = 126

Divide by 15:

k = 126/15

k = 8.4

Therefore, the value of k is 8.4.

Step-by-step explanation: