To determine: Factor a) x2−4x+3 b) 2y2−5y+2 c) 6z2−13z+6

Question

To determine: Factor  a) x2−4x+3  b) 2y2−5y+2  c) 6z2−13z+6

abonirali59 · Accepted Answer

Step 1
a) Factoring Quadratics:
We can get a quadraric by multiplying two first-degree polynomials
For example:

(x + 4) (x - 2)

By using FOIL method

(x + 4) (x - 2) = x (x - 2) + 4 (x - 2)

= x^{2} - 2 x + 4 x - 8

= x^{2} + 2 x - 8

Factoring quadratics needs FOIL backward because factoring is the reverse of multiplication
Given:

x^{2} - 4 x + 3

We have to find integers b and d such that

x^{2} - 4 x + 3 = (x + b) (x + d)

= x^{2} + d x + b x + b d

x^{2} - 4 x + 3 = x^{2} + (b + d) x + b d

Since the constant coefficients on each side of the equation ought to be equal, we must have

b d = 3

(i.e) b and d are factors of 3.
Similarly, the coefficients of x must be the same, so that

b + d = - 4

The following table shows the possibilities

\begin{array}{cc} Factors b, d of 3 & Sum b + d = - 4 \\ 1 \times 3 & 1 + 3 = 4 \\ 1 (- 3) & 1 - 3 = - 2 \\ (- 1) (- 3) & - 1 - 3 = - 4 \end{array}

From the above table, the only factors with product 3 and the sum -4 are -1 and -3
So the correct factorization os

x^{2} - 4 x + 3 = (x - 1) (x - 3)

To check:

(x - 1) (x - 3) = x (x - 3) - 1 (x - 3)

= x^{2} - 3 x - x + 3

= x^{2} - 4 x + 3

Juan Spiller · Accepted Answer

Step 1  b) Given: 2y2−5y+2  We have to find integers a, b, c and d such that  2y2−5y+2=(ay+b)(cy+d)  =acy2+ady+bcy+bd  2y2−55y+2=acy2+(ad+bc)y+bd  Since the coefficient of y2 ought to be same on both sides, we find that ac=2. Likewise, the constant term bd=2. The positive factors of 2 are 2 and 1. Since the midterm is negative, we consider only negative factors of 2. The possibilities are -2 and -1. Now we have to try various arrangements of these factors until we find the one which gives correct coefficient of y.  (2y−2)(y−1)=2y(y−1)−2(y−1)  =2y2−2y−2y+2  =2y2−4y+2  (2y−1)(y−2)=2y(y−2)−1(y−2)  =2y2−4y−y+2  =2y2−5y+2  The last trial gives the correct factorization.

karton · Accepted Answer

Step 1c)Given:6z2−13z+66z2−13z+5=(az+b)(cz+d)=acz2+adz+bcz+bd6z2−13z+6=acz2+(ad+bc)z+bdStep 2(3z−1)(2z−6)=3z(2z−6)−1(2z−6)=6z2−18z−2z+6=6z2−20z+6(2z−3)(3z−2)=2z(3z−2)−3(3z−2)=6z2−4z−9z+6=6z2−13z+6

To determine: Factor a) x^{2}-4x+3 b) 2y^{2}-5y+2 c) 6z^{2}-13z+6

Answered question

Answer & Explanation

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