Write the following expression as a simplified polynomial in standard form: (x-5)^2 + 5(x-5) - 3
Understand the Problem
The question asks to simplify and express a given polynomial expression in standard form. This involves expanding the terms, combining like terms, and arranging the terms in descending order based on their exponents.
Answer
$x^2 - 5x - 3$
Answer for screen readers
$x^2 - 5x - 3$
Steps to Solve
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Expand $(x-5)^2$ Expand the squared term using the formula $(a-b)^2=a^2-2ab+b^2$. $$(x-5)^2 = x^2 - 2(x)(5) + 5^2 = x^2 - 10x + 25$$
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Distribute $5$ in $5(x-5)$ Distribute the $5$ across $(x-5)$. $$5(x-5) = 5x - 25$$
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Substitute the expanded terms back into the original expression Replace $(x-5)^2$ and $5(x-5)$ in the original expression with their expanded forms. $$(x-5)^2 + 5(x-5) - 3 = (x^2 - 10x + 25) + (5x - 25) - 3$$
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Combine like terms Combine the $x^2$ terms, the $x$ terms, and the constant terms. $$x^2 + (-10x + 5x) + (25 - 25 - 3) = x^2 - 5x - 3$$
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Write in Standard Form The simplified expression is already in standard form, which means the terms are arranged in descending order of their exponents.
$x^2 - 5x - 3$
More Information
The standard form of a polynomial is when the terms are arranged from the highest degree to the lowest degree. In this case, the highest degree is 2 ($x^2$), followed by degree 1 ($-5x$), and finally the constant term ($-3$).
Tips
A common mistake is to incorrectly expand $(x-5)^2$. Remember to use the FOIL method or the formula $(a-b)^2 = a^2 - 2ab + b^2$. Another common mistake is to forget to distribute the $5$ properly in $5(x-5)$, or making sign errors when combining the like terms.
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