Question - Factoring Trinomials
Solution:
The given trinomial is $$3x^2 + 17x + 10$$. To factor it we need to find two numbers that multiply to $$3x^2 \cdot 10$$ (the coefficient of $$x^2$$ times the constant term) and add up to $$17x$$ (the coefficient of the middle term).The two numbers that fit this criterion are 2 and 15, since:$$2 \cdot 15 = 30$$, and $$30$$ is the product of the coefficient of $$x^2$$ (3) and the constant term (10).$$2 + 15 = 17$$, which is the coefficient of the middle term.Now we need to split the middle term, $$17x$$, into two terms using the numbers 2 and 15:$$3x^2 + 2x + 15x + 10$$Next, we group terms:$$(3x^2 + 2x) + (15x + 10)$$Factor out the common factor from each group:$$x(3x + 2) + 5(3x + 2)$$We now have a common binomial factor of $$(3x + 2)$$ which we can factor out:$$(3x + 2)(x + 5)$$Therefore, the trinomial $$3x^2 + 17x + 10$$ factors to $$(3x + 2)(x + 5)$$.
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