Final answer:
To factor the polynomial 7x⁴+37x²+10, perform substitution with y=x², factor by grouping the resulting quadratic, and then substitute x² back in to get (7x²+35)(x²+2) as the final factored expression.
Step-by-step explanation:
The question asks to factor a higher degree polynomial, specifically 7x⁴+37x²+10. To factor this trinomial, we treat it as a quadratic in form by using a substitution method where x² is substituted with a variable, let's say 'y'. This gives us 7y²+37y+10. We then look for two numbers that multiply to give (7)(10)=70 and add to give 37. These numbers are 35 and 2. The factored form of the quadratic in y is 7y²+35y+2y+10. Factor by grouping, which gives us y(7y+35)+2(7y+35). We can then factor out the common factor (7y+35), resulting in (7y+35)(y+2). Substituting back for x^2, we get the factored form of the original polynomial: (7x²+35)(x²+2).
When factoring, eliminate terms wherever possible to simplify the algebra and always check the answer to see if it is reasonable. Integer powers like 4³ simply mean 4*4*4. For non-integer powers, a calculator might be used to find the precise value, as it's not as straightforward as multiplication. Still, the power applies to everything inside the parentheses. Remember to multiply both sides by the same factor to keep the equation balanced if necessary.