Match each expression with the correct solution.
Understand the Problem
The prompt provides a series of polynomial subtraction problems and a set of potential solutions. The problem is to match each expression with its correct simplified form. This requires understanding how to combine like terms after distributing the negative sign in the subtraction.
Answer
$(6x^3-2x^2+1)-(8x^2+4) = 6x^3-10x^2-3$ $(8x^2+6x+5x^4)-(2x-7x^4+5) = 12x^4+8x^2+4x-5$ $(4x^4+9x^2)-(2x^4-3x^2+8x-2) = 2x^4+12x^2-8x+2$ $(12x^2-6x-10x^2)-(6x-3x^5-16x^2) = 3x^5+18x^2-12x$ $(3x^4+6x^5-2x^2)-(5x^2-9x^5+3x^4) = 15x^5-7x^2$ $(-x^3+4x^2-9)-(-7x^3+2x^2-13) = 6x^3+2x^2+4$
Answer for screen readers
$(6x^3-2x^2+1)-(8x^2+4) = 6x^3-10x^2-3$
$(8x^2+6x+5x^4)-(2x-7x^4+5) = 12x^4+8x^2+4x-5$
$(4x^4+9x^2)-(2x^4-3x^2+8x-2) = 2x^4+12x^2-8x+2$
$(12x^2-6x-10x^2)-(6x-3x^5-16x^2) = 3x^5+18x^2-12x$
$(3x^4+6x^5-2x^2)-(5x^2-9x^5+3x^4) = 15x^5-7x^2$
$(-x^3+4x^2-9)-(-7x^3+2x^2-13) = 6x^3+2x^2+4$
Steps to Solve
- Simplify $(6x^3-2x^2+1)-(8x^2+4)$
Distribute the negative sign: $6x^3 - 2x^2 + 1 - 8x^2 - 4$ Combine like terms: $6x^3 + (-2x^2 - 8x^2) + (1 - 4)$ $6x^3 - 10x^2 - 3$
- Simplify $(8x^2+6x+5x^4)-(2x-7x^4+5)$
Distribute the negative sign: $8x^2 + 6x + 5x^4 - 2x + 7x^4 - 5$ Combine like terms: $(5x^4 + 7x^4) + 8x^2 + (6x - 2x) - 5$ $12x^4 + 8x^2 + 4x - 5$
- Simplify $(4x^4+9x^2)-(2x^4-3x^2+8x-2)$
Distribute the negative sign: $4x^4 + 9x^2 - 2x^4 + 3x^2 - 8x + 2$ Combine like terms: $(4x^4 - 2x^4) + (9x^2 + 3x^2) - 8x + 2$ $2x^4 + 12x^2 - 8x + 2$
- Simplify $(12x^2-6x-10x^2)-(6x-3x^5-16x^2)$
Distribute the negative sign: $12x^2 - 6x - 10x^2 - 6x + 3x^5 + 16x^2$ Combine like terms: $3x^5 + (12x^2 - 10x^2 + 16x^2) + (-6x - 6x)$ $3x^5 + 18x^2 - 12x$ Rearrange: $3x^5 + 18x^2 - 12x$ Note: None of the answers match this solution, however, $15x^5+6x^2- 12x$ is written incorrectly, and should say $3x^5 + 18x^2 - 12x$
- Simplify $(3x^4+6x^5-2x^2)-(5x^2-9x^5+3x^4)$
Distribute the negative sign: $3x^4 + 6x^5 - 2x^2 - 5x^2 + 9x^5 - 3x^4$ Combine like terms: $(6x^5 + 9x^5) + (3x^4 - 3x^4) + (-2x^2 - 5x^2)$ $15x^5 - 7x^2$
- Simplify $(-x^3+4x^2-9)-(-7x^3+2x^2-13)$
Distribute the negative sign: $-x^3 + 4x^2 - 9 + 7x^3 - 2x^2 + 13$ Combine like terms: $(-x^3 + 7x^3) + (4x^2 - 2x^2) + (-9 + 13)$ $6x^3 + 2x^2 + 4$
$(6x^3-2x^2+1)-(8x^2+4) = 6x^3-10x^2-3$
$(8x^2+6x+5x^4)-(2x-7x^4+5) = 12x^4+8x^2+4x-5$
$(4x^4+9x^2)-(2x^4-3x^2+8x-2) = 2x^4+12x^2-8x+2$
$(12x^2-6x-10x^2)-(6x-3x^5-16x^2) = 3x^5+18x^2-12x$
$(3x^4+6x^5-2x^2)-(5x^2-9x^5+3x^4) = 15x^5-7x^2$
$(-x^3+4x^2-9)-(-7x^3+2x^2-13) = 6x^3+2x^2+4$
More Information
Polynomial subtraction requires careful distribution of the negative sign and accurate combination of like terms.
Tips
A common mistake is not distributing the negative sign correctly across all terms inside the parentheses being subtracted. For example, in $(6x^3-2x^2+1)-(8x^2+4)$, forgetting to apply the negative to both $8x^2$ and $4$ would lead to an incorrect result. Another common mistake is combining unlike terms. For example, adding $x^2$ and $x^3$.
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