Questions: Simplify the expression by combining like terms. [ (-5 x^4+x^3-7 x-6)-(-7 x^3+5 x^2-11) ] (a) 2 x^4+x^3-5 x^2-7 x+5 (b) -5 x^4+7 x^3+6 x^2-7 x-17 (c) -5 x^4+8 x^3+5 x^2-7 x-17 (d) -5 x^4+8 x^3-5 x^2-7 x+5

Solution Steps

To simplify the expression, distribute the negative sign across the terms in the second polynomial and then combine like terms. This involves adding or subtracting the coefficients of terms with the same degree.

To simplify the expression \(\left(-5x^4 + x^3 - 7x - 6\right) - \left(-7x^3 + 5x^2 - 11\right)\), we first distribute the negative sign across the terms in the second polynomial:

\[ -5x^4 + x^3 - 7x - 6 + 7x^3 - 5x^2 + 11 \]

Next, we combine like terms by adding or subtracting the coefficients of terms with the same degree:

• The \(x^4\) term: \(-5x^4\)
• The \(x^3\) terms: \(x^3 + 7x^3 = 8x^3\)
• The \(x^2\) term: \(-5x^2\)
• The \(x\) term: \(-7x\)
• The constant terms: \(-6 + 11 = 5\)

Thus, the simplified expression is:

\[ -5x^4 + 8x^3 - 5x^2 - 7x + 5 \]

Final Answer