1. When the expression $4(x^2-2x+2)-7(x^3-3x+1)$ is fully simplified, what is the sum of the squares of the coefficients of the terms?
2. Find the constant $a$ such that \[(x^2 - 3x + 4)(2x^2 +ax + 7) = 2x^4 -11x^3 +30x^2 -41x +28.\]
+349 1. Distributing the 4 and the -7, we get \(4x^2-8x+8-7x^3+21x-7\). Combining like terms, we get \(-7x^3+4x^2+13x+1\). The coefficients are -7, 4, and 13. The sum of their squares is 49+16+169 = 234.
2. Multiplying the two polynomials (multiply each possible pair of terms and add), we get \(2x^4+ax^3+7x^2-6x^3-3ax^2-21x+8x^2+4ax+28=2x^4+(a-6)x^3+(15-3a)x^2+(4a-21)x+28\)Since the coefficients of the two expressions (this expression and the second expression you gave) have to be equal for the polynomials to be equal, \(a-6=-11\) and \(15-3a=30\) and \(4a-21=-41\). Solving for a, we get a = -5.
+349 1. Distributing the 4 and the -7, we get \(4x^2-8x+8-7x^3+21x-7\). Combining like terms, we get \(-7x^3+4x^2+13x+1\). The coefficients are -7, 4, and 13. The sum of their squares is 49+16+169 = 234.
2. Multiplying the two polynomials (multiply each possible pair of terms and add), we get \(2x^4+ax^3+7x^2-6x^3-3ax^2-21x+8x^2+4ax+28=2x^4+(a-6)x^3+(15-3a)x^2+(4a-21)x+28\)Since the coefficients of the two expressions (this expression and the second expression you gave) have to be equal for the polynomials to be equal, \(a-6=-11\) and \(15-3a=30\) and \(4a-21=-41\). Solving for a, we get a = -5.
