Factoring Trinomials: True / false

Complete the equationUsing quadrilateralsWorded problemsUsing short multiplication formulasHow many solutions exist to a problem?More than one factorizationSolving the equationEquation factorizationTest if the coefficient is different from 1Applying the formulaSolving the problem
Question 1

Determine whether the following statement is true:

\( x^2+16x+64=0 \)

is \( (x+8)(x+8)=0 \)

Question 2

Determine whether the two expressions are corresponding:

\( x^2+3x-18=0 \)

is \( (x+6)(x-3)=0 \)

Question 3

Determine whether the two expressions are corresponding:

\( x^2+x-20=0 \)

is \( (x+2)(x-10)=0 \)

Question 4

Determine whether the given expression is correct:

\( x^2-5x+6=0 \)

is \( (x-5)(x+6) \)

Examples with solutions for Factoring Trinomials: True / false

Exercise #1

Determine whether the following statement is true:

x2+16x+64=0 x^2+16x+64=0

is (x+8)(x+8)=0 (x+8)(x+8)=0

Video Solution

Step-by-Step Solution

Apply trinomial factoring the given expression:

x2+16x+64 x^2+16x+64\\

Let's look for a pair of numbers whose product is the free term in the expression, and their sum is the coefficient of the first-degree term in the expression, meaning two numbers m,n m,\hspace{2pt}n that satisfy:

m⋅n=64m+n=16 m\cdot n=64\\ m+n=16\\ From the first requirement mentioned, namely- from the multiplication, note that the product of the numbers we're looking for must yield a positive result. Therefore we can conclude that both numbers have equal signs, according to the multiplication rules. Remember that 64 has several possible pairs of whole number factors, we won't list all possibilities here, but note that:

64=82=8⋅8 64=8^2=8\cdot8 and- 16=8+8 16=8+8 Continuing, meeting the second requirement mentioned, along with the fact that the signs of the numbers we're looking for are equal to each other will lead us to the conclusion that the only possibility for the two numbers we're looking for is:

{m=8n=8 \begin{cases} m=8\\ n=8\end{cases}

(We can check all other factor pairs of 64 to verify this is the only possibility, but once a suitable option is found - it must be the only one)

Therefore we can factor the given expression to:

x2+16x+64↓(x+8)(x+8) x^2+16x+64\\ \downarrow\\ (x+8)(x+8)

The suggested factorization in the problem is correct.

That is - the correct answer is answer A.

Note:

The given question could also be solved by expanding the parentheses in the suggested expression:

(x+8)(x+8) (x+8)(x+8) (using the extended distribution law or alternatively using the shortened multiplication formula for squared binomial in this case), and checking if indeed we obtain the given expression:

x2+16x+64 x^2+16x+64 , However it's obviously better to try to factor the given expression- for practice purposes.

Answer

True

Exercise #2

Determine whether the two expressions are corresponding:

x2+3x−18=0 x^2+3x-18=0

is (x+6)(x−3)=0 (x+6)(x-3)=0

Video Solution

Step-by-Step Solution

Let's begin by factorizing the given expression using quick trinomial factoring:

x2+3x−18 x^2+3x-18

We'll look for a pair of numbers whose product is the free term in the expression, and whose sum is the coefficient of the first-degree term, meaning two numbers m,n m,\hspace{2pt}n that satisfy:

m⋅n=−18m+n=3 m\cdot n=-18\\ m+n=3\\ From the first requirement mentioned, that is - from the multiplication, we should note that the product of the numbers we're looking for needs to yield a negative result. Therefore we can conclude that the two numbers have different signs, according to multiplication rules. The possible factors of 18 are 2 and 9, 6 and 3, or 18 and 1. Meeting the second requirement mentioned, along with the fact that the numbers we're looking for have different signs leads us to the conclusion that the only possibility for the two numbers we're looking for is:

{m=6n=−3 \begin{cases} m=6\\ n=-3 \end{cases}

Therefore we'll factorize the given expression to:

x2+3x−18↓(x+6)(x−3) x^2+3x-18 \\ \downarrow\\ (x+6)(x-3)

It is clear that the suggested factorization in the problem is correct.

Therefore - the correct answer is answer A.

Note:

The given question could also be solved by expanding the parentheses in the suggested expression:

(x+6)(x−3) (x+6)(x-3) (using the expanded distributive law), and checking if indeed we obtain the given expression:

x2+3x−18 x^2+3x-18 , However it is of course preferable to try to factorize the given expression - for practice purposes.

Answer

True

Exercise #3

Determine whether the two expressions are corresponding:

x2+x−20=0 x^2+x-20=0

is (x+2)(x−10)=0 (x+2)(x-10)=0

Video Solution

Step-by-Step Solution

Let's ascertain whether the following expression can be factorized using quick trinomial factoring:

x2+x−20 x^2+x-20

We'll begin by looking for a pair of numbers whose product is the free term in the expression, and whose sum is the coefficient of the first power term in the expression . Meaning two numbers m,n m,\hspace{2pt}n that satisfy the following expression:

m⋅n=−20m+n=1 m\cdot n=-20\\ m+n=1\\ From the first requirement mentioned, namely - from the multiplication, we should note that the product of the numbers we're looking for needs to yield a negative result. Therefore we can conclude that the two numbers have different signs, according to the multiplication rules. The possible factors of 20 are 2 and 10, 4 and 5, or 20 and 1, fulfilling the second requirement mentioned, along with the fact that the signs of the numbers we're looking for are different from each other lead us to the conclusion that the only possibility for the two numbers we're looking for is:

{m=5n=−4 \begin{cases} m=5\\ n=-4 \end{cases}

Therefore we'll factorize the given expression to:

x2+x−20↓(x+5)(x−4) x^2+x-20 \\ \downarrow\\ (x+5)(x-4)

Evidently the suggested factorization in the problem is incorrect.

Therefore - the correct answer is answer B.

Note:

The given question could also be solved by expanding the parentheses in the suggested expression:

(x+2)(x−10) (x+2)(x-10) (using the expanded distributive property), and checking if it indeed equals the given expression:

x2+x−20 x^2+x-20 , It is of course preferable to try to factorize the given expression - for practice purposes.

Answer

Not true

Exercise #4

Determine whether the given expression is correct:

x2−5x+6=0 x^2-5x+6=0

is (x−5)(x+6) (x-5)(x+6)

Video Solution

Step-by-Step Solution

Apply quick trinomial factoring to try and factor the given expression:

x2−5x+6 x^2-5x+6

Look for a pair of numbers whose product is the free term in the expression, and their sum is the coefficient of the first power term in the expression, meaning two numbers m,n m,\hspace{2pt}n that satisfy:

m⋅n=6m+n=−5 m\cdot n=6\\ m+n=-5\\ From the first requirement mentioned, namely- from the multiplication, we should note that the product of the numbers we're looking for needs to yield a positive result. Therefore we can conclude that both numbers have the same signs, according to multiplication rules. Remember that the possible factors of 6 are 2 and 3 or 6 and 1. This satisfies the second requirement mentioned. Together with the fact that the signs of the numbers we're looking for are equal to each other leads us to the conclusion that the only possibility for the two numbers we're looking for is:

{m=−3n=−2 \begin{cases} m=-3\\ n=-2 \end{cases}

Therefore we proceed to factor the given expression to:

x2−5x+6↓(x−3)(x−2) x^2-5x+6 \\ \downarrow\\ (x-3)(x-2)

Clearly the factorization suggested in the problem is incorrect.

Therefore- the correct answer is answer B.

Note:

The given question could also be solved by expanding the parentheses in the suggested expression:

(x−5)(x+6) (x-5)(x+6) (using the expanded distributive property), and checking if indeed we obtain the given expression:

x2−5x+6 x^2-5x+6 , However it is of course preferable to try to factor the given expression- for practice purposes.

Answer

Not true