Solved on Feb 10, 2024

Find $a$ and $b$ such that $P(x) = (2a - b)x^2 + (b - 6)x + a - 3$ is the zero polynomial.

STEP 1

Assumptions
1. The polynomial $P(x)$ is identically null, which means that $P(x) = 0$ for all values of $x$ .
2. The coefficients of $x^2$ , $x$ , and the constant term in $P(x)$ must all be zero for the polynomial to be identically null.

STEP 2

Write down the polynomial $P(x)$ and equate each coefficient to zero, since $P(x)$ is identically null.
$P(x)=(2a-b)x^2 + (b-6)x + (a-3)$

STEP 3

Equate the coefficient of $x^2$ to zero.
$2a - b = 0$

STEP 4

Equate the coefficient of $x$ to zero.
$b - 6 = 0$

STEP 5

Equate the constant term to zero.
$a - 3 = 0$

STEP 6

Solve the equation from STEP_4 to find the value of $b$ .
$b - 6 = 0 \Rightarrow b = 6$

STEP 7

Solve the equation from STEP_5 to find the value of $a$ .
$a - 3 = 0 \Rightarrow a = 3$

STEP 8

Substitute the value of $b$ into the equation from STEP_3 to find the value of $a$ .
$2a - b = 0 \Rightarrow 2a - 6 = 0$

STEP 9

Solve the equation from STEP_8.
$2a - 6 = 0 \Rightarrow 2a = 6 \Rightarrow a = 3$

STEP 10

Verify that the values of $a$ and $b$ satisfy all the equations.
For $2a - b = 0$ :
$2(3) - 6 = 0 \Rightarrow 6 - 6 = 0 \Rightarrow 0 = 0$
For $b - 6 = 0$ :
$6 - 6 = 0 \Rightarrow 0 = 0$
For $a - 3 = 0$ :
$3 - 3 = 0 \Rightarrow 0 = 0$

STEP 11

Since all the equations are satisfied, the values of $a$ and $b$ are correct. Therefore, the values of $a$ and $b$ that make the polynomial $P(x)$ identically null are:
$a = 3$ $b = 6$

Was this helpful?

Find $a$ and $b$ such that $P(x) = (2a - b)x^2 + (b - 6)x + a - 3$ is the zero polynomial.

STEP 1

Assumptions
1. The polynomial $P(x)$ is identically null, which means that $P(x) = 0$ for all values of $x$ .
2. The coefficients of $x^2$ , $x$ , and the constant term in $P(x)$ must all be zero for the polynomial to be identically null.

STEP 2

Write down the polynomial $P(x)$ and equate each coefficient to zero, since $P(x)$ is identically null.
$P(x)=(2a-b)x^2 + (b-6)x + (a-3)$

STEP 3

Equate the coefficient of $x^2$ to zero.
$2a - b = 0$

STEP 4

Equate the coefficient of $x$ to zero.
$b - 6 = 0$

STEP 5

Equate the constant term to zero.
$a - 3 = 0$

STEP 6

Solve the equation from STEP_4 to find the value of $b$ .
$b - 6 = 0 \Rightarrow b = 6$

STEP 7

Solve the equation from STEP_5 to find the value of $a$ .
$a - 3 = 0 \Rightarrow a = 3$

STEP 8

Substitute the value of $b$ into the equation from STEP_3 to find the value of $a$ .
$2a - b = 0 \Rightarrow 2a - 6 = 0$

STEP 9

Solve the equation from STEP_8.
$2a - 6 = 0 \Rightarrow 2a = 6 \Rightarrow a = 3$

STEP 10

Verify that the values of $a$ and $b$ satisfy all the equations.
For $2a - b = 0$ :
$2(3) - 6 = 0 \Rightarrow 6 - 6 = 0 \Rightarrow 0 = 0$
For $b - 6 = 0$ :
$6 - 6 = 0 \Rightarrow 0 = 0$
For $a - 3 = 0$ :
$3 - 3 = 0 \Rightarrow 0 = 0$

STEP 11

Since all the equations are satisfied, the values of $a$ and $b$ are correct. Therefore, the values of $a$ and $b$ that make the polynomial $P(x)$ identically null are:
$a = 3$ $b = 6$

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find aaa and bbb such that P(x)=(2a−b)x2+(b−6)x+a−3P(x) = (2a - b)x^2 + (b - 6)x + a - 3P(x)=(2a−b)x2+(b−6)x+a−3 is the zero polynomial.

STEP 1

Assumptions1. The polynomial P(x)P(x)P(x) is identically null, which means that P(x)=0P(x) = 0P(x)=0 for all values of xxx.2. The coefficients of x2x^2x2, xxx, and the constant term in P(x)P(x)P(x) must all be zero for the polynomial to be identically null.

STEP 2

Write down the polynomial P(x)P(x)P(x) and equate each coefficient to zero, since P(x)P(x)P(x) is identically null.P(x)=(2a−b)x2+(b−6)x+(a−3)P(x)=(2a-b)x^2 + (b-6)x + (a-3)P(x)=(2a−b)x2+(b−6)x+(a−3)

STEP 3

Equate the coefficient of x2x^2x2 to zero.2a−b=02a - b = 02a−b=0

STEP 4

Equate the coefficient of xxx to zero.b−6=0b - 6 = 0b−6=0

STEP 5

Equate the constant term to zero.a−3=0a - 3 = 0a−3=0

STEP 6

Solve the equation from STEP_4 to find the value of bbb.b−6=0⇒b=6b - 6 = 0 \Rightarrow b = 6b−6=0⇒b=6

STEP 7

Solve the equation from STEP_5 to find the value of aaa.a−3=0⇒a=3a - 3 = 0 \Rightarrow a = 3a−3=0⇒a=3

STEP 8

Substitute the value of bbb into the equation from STEP_3 to find the value of aaa.2a−b=0⇒2a−6=02a - b = 0 \Rightarrow 2a - 6 = 02a−b=0⇒2a−6=0

STEP 9

Solve the equation from STEP_8.2a−6=0⇒2a=6⇒a=32a - 6 = 0 \Rightarrow 2a = 6 \Rightarrow a = 32a−6=0⇒2a=6⇒a=3

STEP 10

STEP 11

Since all the equations are satisfied, the values of aaa and bbb are correct. Therefore, the values of aaa and bbb that make the polynomial P(x)P(x)P(x) identically null are:a=3a = 3a=3 b=6b = 6b=6

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find $a$ and $b$ such that $P(x) = (2a - b)x^2 + (b - 6)x + a - 3$ is the zero polynomial.

Assumptions
1. The polynomial $P(x)$ is identically null, which means that $P(x) = 0$ for all values of $x$ .
2. The coefficients of $x^2$ , $x$ , and the constant term in $P(x)$ must all be zero for the polynomial to be identically null.

Write down the polynomial $P(x)$ and equate each coefficient to zero, since $P(x)$ is identically null.
$P(x)=(2a-b)x^2 + (b-6)x + (a-3)$

Equate the coefficient of $x^2$ to zero.
$2a - b = 0$

Equate the coefficient of $x$ to zero.
$b - 6 = 0$

Equate the constant term to zero.
$a - 3 = 0$

Solve the equation from STEP_4 to find the value of $b$ .
$b - 6 = 0 \Rightarrow b = 6$

Solve the equation from STEP_5 to find the value of $a$ .
$a - 3 = 0 \Rightarrow a = 3$

Substitute the value of $b$ into the equation from STEP_3 to find the value of $a$ .
$2a - b = 0 \Rightarrow 2a - 6 = 0$

Solve the equation from STEP_8.
$2a - 6 = 0 \Rightarrow 2a = 6 \Rightarrow a = 3$

Since all the equations are satisfied, the values of $a$ and $b$ are correct. Therefore, the values of $a$ and $b$ that make the polynomial $P(x)$ identically null are:
$a = 3$ $b = 6$