Perform the indicated operations, write the resulting polynomial in standard form, and state its degree: (5x^2 - 7x - 1) + (3x^2 - 5x + 8) - (x^2 - 5x - 2).

Step-by-step solution

1. 1

   Deal with the subtraction sign before combining anything. The minus in front of the third bracket multiplies every term inside it, including the $-2$. Forgetting the last sign is the single most common slip in this problem, so rewrite the whole expression as a pure addition first:

   $\left(5x^{2}-7x-1\right)+\left(3x^{2}-5x+8\right)-x^{2}+5x+2$

   Note that $-(-5x)=+5x$ and $-(-2)=+2$.

2. 2

   Collect the $x^{2}$ terms. Like terms are those with the same power of $x$, and only they may be combined:

   $5x^{2}+3x^{2}-x^{2}=7x^{2}$

3. 3

   Collect the $x$ terms. Two negatives and one positive here:

   $-7x-5x+5x=-7x$

   The $-5x$ and $+5x$ cancel, leaving just the $-7x$ from the first polynomial.

4. 4

   Collect the constants.

   $-1+8+2=9$

5. 5

   Write standard form and read off the degree. Standard form means descending powers of $x$:

   $7x^{2}-7x+9$

   The highest power present is $x^{2}$, so the degree is $2$.

6. 6

   Check by substituting test values. At $x=0$ the original expression is $(-1)+(8)-(-2)=9$ and the result gives $9$. At $x=2.3$ both sides evaluate to $29.93$, and at $x=-1.7$ both give $41.13$. Agreement at three separate values of $x$ confirms the coefficients, since two quadratics matching at three points are identical.