Sure, let's solve the problem step-by-step:
1. Identify the given polynomials:
- Initial polynomial: [tex]\( 7x^2 - 5x + 1 \)[/tex]
- Target polynomial: [tex]\( -3x^2 + 4x + 7 \)[/tex]
2. Set up the problem to find the polynomial that needs to be added:
- We need to determine the polynomial [tex]\( ax^2 + bx + c \)[/tex] that, when added to [tex]\( 7x^2 - 5x + 1 \)[/tex], results in [tex]\( -3x^2 + 4x + 7 \)[/tex].
3. Set up the equation for the sum:
[tex]\[ (7x^2 - 5x + 1) + (ax^2 + bx + c) = -3x^2 + 4x + 7 \][/tex]
4. Compare coefficients for [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and the constant term:
- For [tex]\( x^2 \)[/tex] term:
[tex]\[ 7 + a = -3 \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ a = -3 - 7 = -10 \][/tex]
- For [tex]\( x \)[/tex] term:
[tex]\[ -5 + b = 4 \][/tex]
Solving for [tex]\( b \)[/tex]:
[tex]\[ b = 4 + 5 = 9 \][/tex]
- For the constant term:
[tex]\[ 1 + c = 7 \][/tex]
Solving for [tex]\( c \)[/tex]:
[tex]\[ c = 7 - 1 = 6 \][/tex]
5. Combine the coefficients:
Therefore, the polynomial that should be added is:
[tex]\[ -10x^2 + 9x + 6 \][/tex]
Thus, [tex]\( -10x^2 + 9x + 6 \)[/tex] should be added to [tex]\( 7x^2 - 5x + 1 \)[/tex] to obtain [tex]\( -3x^2 + 4x + 7 \)[/tex].
