Math Problem Statement

(ax + 1)(x - 3)(x + b) = 2x^3 - 3x^2 - 8x - 3. Find the value of a and b?

Solution

The given expression is:

(ax+1)(x3)(x+b)=2x33x28x3(ax + 1)(x - 3)(x + b) = 2x^3 - 3x^2 - 8x - 3

We need to find the values of aa and bb.

Step 1: Expand the left-hand side

We'll first expand (ax+1)(x3)(x+b)(ax + 1)(x - 3)(x + b).

Start by expanding (x3)(x+b)(x - 3)(x + b): (x3)(x+b)=x2+bx3x3b=x2+(b3)x3b(x - 3)(x + b) = x^2 + bx - 3x - 3b = x^2 + (b - 3)x - 3b

Now, multiply this by (ax+1)(ax + 1): (ax+1)(x2+(b3)x3b)=ax(x2+(b3)x3b)+1(x2+(b3)x3b)(ax + 1)(x^2 + (b - 3)x - 3b) = ax(x^2 + (b - 3)x - 3b) + 1(x^2 + (b - 3)x - 3b) =ax3+a(b3)x23abx+x2+(b3)x3b= ax^3 + a(b - 3)x^2 - 3abx + x^2 + (b - 3)x - 3b

Step 2: Collect like terms

Group the terms by powers of xx: =ax3+(a(b3)+1)x2+(3ab+b3)x3b= ax^3 + (a(b - 3) + 1)x^2 + (-3ab + b - 3)x - 3b

Step 3: Compare coefficients with the right-hand side

Now, compare this with the right-hand side 2x33x28x32x^3 - 3x^2 - 8x - 3.

  • Coefficient of x3x^3: a=2a = 2
  • Coefficient of x2x^2: a(b3)+1=3a(b - 3) + 1 = -3
  • Coefficient of xx: 3ab+b3=8-3ab + b - 3 = -8
  • Constant term: 3b=3-3b = -3

Step 4: Solve for aa and bb

  1. From the constant term equation: 3b=3b=1-3b = -3 \quad \Rightarrow \quad b = 1

  2. Substitute b=1b = 1 into the equation for x2x^2: a(13)+1=3a(2)+1=32a=4a=2a(1 - 3) + 1 = -3 \quad \Rightarrow \quad a(-2) + 1 = -3 \quad \Rightarrow \quad -2a = -4 \quad \Rightarrow \quad a = 2

  3. Verify with the equation for xx: 3ab+b3=83(2)(1)+13=8-3ab + b - 3 = -8 \quad \Rightarrow \quad -3(2)(1) + 1 - 3 = -8 6+13=8(True)-6 + 1 - 3 = -8 \quad \text{(True)}

Thus, the values of aa and bb are: a=2,b=1a = 2, \quad b = 1

Would you like further details or have any questions?

Related Questions:

  1. How do we expand binomials in general?
  2. What are other ways to solve for unknowns in polynomial identities?
  3. Can this problem be solved using synthetic division?
  4. How does comparing coefficients help in solving polynomial equations?
  5. What are some typical uses of this method in algebra?

Tip:

When expanding products of polynomials, always group terms by powers of xx for easier comparison with the target expression.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Expansion
Comparing Coefficients

Formulas

(ax + 1)(x - 3)(x + b)
Polynomial expansion

Theorems

Polynomial identities
Comparing coefficients

Suitable Grade Level

Grades 9-12