Question

simplify each expression.

1. $(8p - 3p^{4}+8)-(3p^{4}-7 + p)$
2. $(2x^{4}+6 - 7x^{3})-(x^{4}+7x^{3}+1)$
3. $(r^{2}+3r^{4}-6r^{3})-(4r^{2}+2r^{4}-7r^{3})$
4. $(8 - n^{2}-6n^{4})-(1 + n^{2}-4n^{4})$
5. $(8b^{2}-b - 4b^{3})+(7b - 4b^{3}-2)$
6. $(4r^{3}+r^{4}-2r)+(8r^{4}+3r^{3}+8r)$
7. $(3x - 2x^{2}-5x^{4})-(8x - 8x^{2}+8x^{4})$
8. $(3p - 2p^{3}-p^{2})-(8p^{4}+7p^{3}+4p^{2})$
9. $(7n^{3}+5 - 6n^{4})+(2 - 5n^{2}+2n^{3})

Explanation:

Step1: Remove parentheses

For example, in the first expression \((8p - 3p^{4}+8)-(3p^{4}-7 + p)\), we get \(8p - 3p^{4}+8 - 3p^{4}+7 - p\) by distributing the negative sign in the second - parentheses.

Step2: Combine like - terms

Combining the \(p^{4}\) terms \(-3p^{4}-3p^{4}=-6p^{4}\), the \(p\) terms \(8p - p = 7p\), and the constant terms \(8 + 7=15\). So the simplified form is \(-6p^{4}+7p + 15\).

We will follow the same two - step process for all expressions:

1. \((8p - 3p^{4}+8)-(3p^{4}-7 + p)=8p - 3p^{4}+8 - 3p^{4}+7 - p=-6p^{4}+7p + 15\)
2. \((2x^{4}+6 - 7x^{3})-(x^{4}+7x^{3}+1)=2x^{4}+6 - 7x^{3}-x^{4}-7x^{3}-1=x^{4}-14x^{3}+5\)
3. \((r^{2}+3r^{4}-6r^{3})-(4r^{2}+2r^{4}-7r^{3})=r^{2}+3r^{4}-6r^{3}-4r^{2}-2r^{4}+7r^{3}=r^{4}+r^{3}-3r^{2}\)
4. \((8 - n^{2}-6n^{4})-(1 + n^{2}-4n^{4})=8 - n^{2}-6n^{4}-1 - n^{2}+4n^{4}=-2n^{2}-2n^{4}+7\)
5. \((8b^{2}-b - 4b^{3})+(7b - 4b^{3}-2)=8b^{2}-b - 4b^{3}+7b - 4b^{3}-2=8b^{2}+6b - 8b^{3}-2\)
6. \((4r^{3}+r^{4}-2r)+(8r^{4}+3r^{3}+8r)=4r^{3}+r^{4}-2r+8r^{4}+3r^{3}+8r=9r^{4}+7r^{3}+6r\)
7. \((3x - 2x^{2}-5x^{4})-(8x - 8x^{2}+8x^{4})=3x - 2x^{2}-5x^{4}-8x + 8x^{2}-8x^{4}=-13x^{4}+6x^{2}-5x\)
8. \((3p - 2p^{3}-p^{2})-(8p^{4}+7p^{3}+4p^{2})=3p - 2p^{3}-p^{2}-8p^{4}-7p^{3}-4p^{2}=-8p^{4}-9p^{3}-5p^{2}+3p\)
9. \((7n^{3}+5 - 6n^{4})+(2 - 5n^{2}+2n^{3})=-6n^{4}+(7n^{3}+2n^{3})-5n^{2}+(5 + 2)=-6n^{4}+9n^{3}-5n^{2}+7\)

Answer:

1. \(-6p^{4}+7p + 15\)
2. \(x^{4}-14x^{3}+5\)
3. \(r^{4}+r^{3}-3r^{2}\)
4. \(-2n^{2}-2n^{4}+7\)
5. \(8b^{2}+6b - 8b^{3}-2\)
6. \(9r^{4}+7r^{3}+6r\)
7. \(-13x^{4}+6x^{2}-5x\)
8. \(-8p^{4}-9p^{3}-5p^{2}+3p\)
9. \(-6n^{4}+9n^{3}-5n^{2}+7\)