Question

1. factor: $10x^2 - 7x + 1$

Explanation:

Step1: Identify coefficients

For quadratic \(10x^2 - 7x + 1\), \(a = 10\), \(b=-7\), \(c = 1\).

Step2: Find two numbers

Find \(m\) and \(n\) such that \(m\times n=a\times c = 10\times1 = 10\) and \(m + n=b=-7\). The numbers are \(-5\) and \(-2\) (since \((-5)\times(-2)=10\) and \(-5+(-2)=-7\)).

Step3: Split middle term

Rewrite the middle term: \(10x^2-5x - 2x + 1\).

Step4: Group and factor

Group as \((10x^2 - 5x)+(-2x + 1)\). Factor out \(5x\) from first group and \(-1\) from second: \(5x(2x - 1)-1(2x - 1)\).

Step5: Factor out common binomial

Factor out \((2x - 1)\): \((5x - 1)(2x - 1)\).

Answer:

\((5x - 1)(2x - 1)\)