Fill in the blanks to complete the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference.

Question

Fill in the blanks to complete the polynomial equation that shows the subtraction of the second expression from the first and the resulting difference. 
9x^2+_x +13
-2x^2+12x+_
_x^2+5x+2

GPT-4o mini · Answer

To complete the polynomial equation given, we need to find values for the blanks such that:

$(9 x^{2} +_{x} + 13) - (2 x^{2} + 12 x +_{)} =_{x}^{2} + 5 x + 2$

Let's denote the missing terms with variables for clarity:

Let the first blank in the first polynomial be $a$ .
Let the second blank in the second polynomial be $b$ .
Let the blank in the resulting polynomial be $c$ .

This gives us:

$(9 x^{2} + a x + 13) - (2 x^{2} + 12 x + b) = c x^{2} + 5 x + 2$

Now, we distribute the negative sign through the second polynomial:

$9 x^{2} + a x + 13 - 2 x^{2} - 12 x - b = c x^{2} + 5 x + 2$

Combining like terms on the left side results in:

$(9 x^{2} - 2 x^{2}) + (a x - 12 x) + (13 - b) = c x^{2} + 5 x + 2$

This simplifies to:

$(7 x^{2} + (a - 12) x + (13 - b)) = c x^{2} + 5 x + 2$

From this equation, we can set up the following for the coefficients:

From the $x^{2}$ terms: $7 = c$
From the $x$ terms: $a - 12 = 5$
From the constant terms: $13 - b = 2$

Now we solve each of these equations:

Since $7 = c$ , we have $c = 7$ .
For $a - 12 = 5$ : $a = 5 + 12 = 17$
For $13 - b = 2$ : $b = 13 - 2 = 11$

Thus, we find:

The first blank (coefficient of $x$ in the first polynomial) is $17$ .
The second blank (constant term of the second polynomial) is $11$ .
The blank in the resulting polynomial (coefficient of $x^{2}$ ) is $7$ .

Putting it all together, the complete equation becomes:

$9 x^{2} + 17 x + 13 - (2 x^{2} + 12 x + 11) = 7 x^{2} + 5 x + 2$

In summary:

The first blank is 17.
The second blank is 11.
The third blank is 7.