Factor the following expression completely. x^(4)+x^(3)-42x^(2)-x^(2)-x+42 Answer:

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Factor the following expression completely.  x^(4)+x^(3)-42x^(2)-x^(2)-x+42 Answer:

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Combine Like Terms: First, combine like terms in the expression x^4 + x^3 - 42x^2 - x^2 - x + 42. x^4 + x^3 - 42x^2 - x^2 can be combined to x^4 + x^3 - 43x^2. So the expression becomes x^4 + x^3 - 43x^2 - x + 42.Look for Common Factors: Next, look for common factors in groups of terms. We can group the first three terms and the last two terms separately. Grouping gives us (x^4 + x^3 - 43x^2) - (x - 42). Notice that x^2 is a common factor in the first group.Factor Out Common Factor: Factor out the common factor of x^2 from the first group. This gives us x^2(x^2 + x - 43) - (x - 42).Check for Further Factoring: Now, we need to check if x^2 + x - 43 can be factored further. However, it does not factor nicely, so we leave it as is.Examine Second Group: Next, we look at the second group (x - 42) and realize there is no common factor to take out, and it cannot be factored further.Check for Common Factor: Now, we need to check if there is a common factor between x^2(x^2 + x - 43) and (x - 42) that we might have missed. Upon closer inspection, we see that there is no common factor, and we cannot factor by grouping.Explore Other Factoring Techniques: Since we cannot factor by grouping, we need to look for other factoring techniques. One such technique is to look for patterns or to use the rational root theorem or synthetic division to find the roots of the polynomial.Use Rational Root Theorem: Let's try to find the roots of the polynomial by using the rational root theorem. The possible rational roots are the factors of the constant term (42) divided by the factors of the leading coefficient (1). The factors of 42 are ±1, ±2, ±3, ±6, ±7, ±14, ±21, ±42.Test Possible Rational Roots: We test the possible rational roots in the polynomial x^4 + x^3 - 43x^2 - x + 42 to see if any of them are actual roots. After testing, we find that x = -1 is a root because substituting -1 into the polynomial gives 0.Identify Root and Factor: Since x = -1 is a root, (x + 1) is a factor of the polynomial. We can use synthetic division or long division to divide the polynomial by (x + 1) to find the other factors.Perform Synthetic Division: Performing the division, we get the quotient x^3 - 42x^2 + x - 42.

Factor the following expression completely. $\newline$ $x^{4}+x^{3}-42 x^{2}-x^{2}-x+42$ $\newline$ Answer:

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Factor the following expression completely. $\newline$ $x^{4}+x^{3}-42 x^{2}-x^{2}-x+42$ $\newline$ Answer: