Find the constant after subtracting from $x^{4} + 4 x^{2} - 3 x + 7$ to get $3 x^{3} - x^{2} + 2 x + 1$ ?

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Solution

We find the subtraction of the given polynomials $(x^{4} + 4 x^{2} - 3 x + 7) - (3 x^{3} - x^{2} + 2 x + 1)$ by combining the like terms as shown below:

$(x^{4} + 4 x^{2} - 3 x + 7) - (3 x^{3} - x^{2} + 2 x + 1) = x^{4} - 3 x^{3} + (4 x^{2} + x^{2}) + (- 3 x - 2 x) + (7 - 1) = x^{4} - 3 x^{3} + x^{2} (4 + 1) + x (- 3 - 2) + 6 = x^{4} - 3 x^{3} + 5 x^{2} - 5 x + 6$

We know that the constant term of a polynomial is the term of degree $0$ ; it is the term in which the variable does not appear.

In the resulting polynomial $x^{4} - 3 x^{3} + 5 x^{2} - 5 x + 6$ , the constant term is $6$ because there is no variable.

Hence, the constant term is $6$ .

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