Math Problem Statement

Identify the leading term of the polynomial f that is given by f(x) = (4x + 10)^6 * (15x - 11)^5 * (-8x - 7)^4 * (8x + 5)^2 * (-12x - 1)^1.

Solution

To identify the leading term of the given polynomial function f(x)f(x), we need to focus on the highest powers of xx from each factor. The general form of the function is:

f(x)=(4x+10)6(15x11)5(8x7)4(8x+5)2(12x1)1f(x) = (4x + 10)^6 \cdot (15x - 11)^5 \cdot (-8x - 7)^4 \cdot (8x + 5)^2 \cdot (-12x - 1)^1

Steps to find the leading term:

  1. Extract the highest power of xx from each factor:

    • (4x+10)6(4x + 10)^6 has a leading term of (4x)6=46x6=4096x6(4x)^6 = 4^6 x^6 = 4096x^6
    • (15x11)5(15x - 11)^5 has a leading term of (15x)5=155x5=759375x5(15x)^5 = 15^5 x^5 = 759375x^5
    • (8x7)4(-8x - 7)^4 has a leading term of (8x)4=(8)4x4=4096x4(-8x)^4 = (-8)^4 x^4 = 4096x^4
    • (8x+5)2(8x + 5)^2 has a leading term of (8x)2=82x2=64x2(8x)^2 = 8^2 x^2 = 64x^2
    • (12x1)1(-12x - 1)^1 has a leading term of (12x)1=12x(-12x)^1 = -12x
  2. Multiply the leading coefficients: Multiply the leading terms of each factor: 4096x6759375x54096x464x2(12x)4096x^6 \cdot 759375x^5 \cdot 4096x^4 \cdot 64x^2 \cdot (-12x)

    The coefficients: 4096759375409664(12)4096 \cdot 759375 \cdot 4096 \cdot 64 \cdot (-12)

    The powers of xx: x6+5+4+2+1=x18x^{6 + 5 + 4 + 2 + 1} = x^{18}

Now I'll calculate the product of the coefficients.The product of the coefficients is 9784472371200000-9784472371200000. Therefore, the leading term of the polynomial is:

9784472371200000x18-9784472371200000 \, x^{18}

Let me know if you want more details or have questions!

Here are 5 related questions to expand your understanding:

  1. How do we determine the degree of a polynomial from its factors?
  2. What is the significance of the leading term in the context of polynomial functions?
  3. How would the leading term change if one of the coefficients in the factors were different?
  4. How do leading terms affect the end behavior of a polynomial?
  5. Can a polynomial have more than one leading term?

Tip: Always check the sign of the leading coefficient, as it affects the polynomial's graph orientation!

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Math Problem Analysis

Mathematical Concepts

Polynomials
Leading Terms
Exponentiation

Formulas

Leading term of a product of polynomials is the product of the leading terms of each factor
(ax + b)^n has leading term (a^n)x^n

Theorems

Degree of a polynomial is the sum of the degrees of its factors

Suitable Grade Level

Grades 10-12