Question

How do you factor the trinomial \[{m^2} - 15m + 50\] ?

Answer 1

Hint: The definition of a trinomial is a math equation that has three terms which are connected by plus or minus notations. The given expression is a polynomial of degree 2. Instead of ‘x’ as a variable we have ‘m’ as a variable. We can solve this by using factorization methods or by using quadratic formulas. We use quadratic formula if factorization fails. We know that a polynomial equation has exactly as many roots as its degree.

Complete step-by-step answer:
The degree of the equation \[{m^2} - 15m + 50\] is 2, so the number of roots of the given equation is 2.
On comparing the given equation with the standard quadratic equation \[A{m^2} + Bm + C = 0\] . Where ‘A’ and ‘B’ are coefficients of \[{m^2}\] and coefficient of ‘m’ respectively.
We have \[A = 1\] , \[B = - 15\] and \[C = 50\] .
The standard form of the factorization of quadratic equation is \[A{m^2} + {B_1}m + {B_2}m + C = 0\] , which satisfies the condition \[{B_1} \times {B_2} = A \times C\] and \[{B_1} + {B_2} = B\] .We can write the given equation as \[{m^2} - 5m - 10m + 50\] , where \[{B_1} = - 5\] and \[{B_2} = - 10\] .
Also \[{B_1} \times {B_2} = ( - 5) \times ( - 10) = 50(A \times C)\] and \[{B_1} + {B_2} = ( - 5) + ( - 10) = - 15(B)\] .
 \[ \Rightarrow {m^2} - 15m + 50 = {m^2} - 5m - 10m + 50\]
 \[ = {m^2} - 5m - 10m + 50\]
In the first two terms we take ‘m’ as common and in the remaining term we take -10 as common,
 \[ = m(m - 5) - 10(m - 5)\]
Again taking \[(m - 5)\] as common we have,
 \[ = (m - 5)(m - 10)\]
Hence, the factors of \[{m^2} - 15m + 50\] are \[(m - 5)\] and \[(m - 10)\] .
So, the correct answer is “ \[(m - 5)\] and \[(m - 10)\] ”.

Note: We can find the zeros or roots of the given quadratic expression by equating the obtained factors to zero. That is,
 \[(m - 5)(m - 10) = 0\] .
By zero multiplication property we have,
 \[(m - 5) = 0\] and \[(m - 10) = 0\] .
 \[ \Rightarrow m = 5\] and \[m = 10\] .
These are the roots of the given problem.
In various fields of mathematics require the point at which the value of a polynomial is zero, those values are called the factors/solution/zeros of the given polynomial. On the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x-intercepts.