Question

The graph shows the system ​ {y=2x+1
{ y=x^2+1 .​ Which ordered pairs are solutions of the system? ​
A. (0, 1) ​ and ​ (2, 5) ​ ​
B. (0, 0) ​ and ​ (2, 5) ​
C. (1, 0) and (4, 2) ​
D. (0, 1) and ​ (1, 4) ​ ​

2. What is the product of 1/2x - 1/4 and 5x^2 - 2x + 6 and equal to he product of 1/4x - 1/2 and 5x^2 - 2x + 6 ? Write your answer in standard form.

Answer 1

y = 2x + 1
y = x²+1
reemplazando
2x+1 = x²+1
ordenando
0=x²-2x
donde
0 = x (x-2)
resolviendo
0=x      v     x-2=0
0 =x            x= 2

si x=0  entonces y = 1    par (0,1)
si x=2  entonces y= 5     par (2,5)

respuesta  A

Answer 2

1) The ordered pairs $(0, 1)$ and $(2,5)$ are solutions of the system.

2) The product of $\frac{1}{2}\cdot x - \frac{1}{4}$ and $5\cdot x^{2}-2\cdot x + 6$ is $\frac{5}{2}\cdot x^{5}-\frac{5}{4}\cdot x^{4}-x^{2}+\frac{7}{2}\cdot x-\frac{3}{2}$.

How to solve a system of two polynomic equations

1) In this question, we must solve for $x$ and $y$ the following system of equations:

$y = 2\cdot x + 1$  (1)

$y=x^{2}+1$  (2)

By (1) and (2):

$2\cdot x + 1 = x^{2}+1$

$x^{2}-2\cdot x = 0$

$x\cdot (x-2) = 0$

The solutions for $x$ are 0 and 2, respectively.

Lastly, we find the solutions for $y$ by (1):

x = 0

$y = 2\cdot 0 + 1$

$y = 1$

x = 2

$y = 2\cdot 2 +1$

$y = 5$

The solutions for $y$ are 1 and 5, respectively.

The ordered pairs $(0, 1)$ and $(2,5)$ are solutions of the system. $\blacksquare$

How to find resulting polynomial by operations between polynomials

2) In this case, we must determine the product of two polynomials by algebraic means:

1. $\left(\frac{1}{2}\cdot x-\frac{1}{4} \right)\cdot (5\cdot x^{2}-2\cdot x + 6)$     Given
2. $\left(\frac{1}{2}\cdot x - \frac{1}{4} \right)\cdot (5\cdot x^{4})+\left(\frac{1}{2}\cdot x - \frac{1}{4} \right)\cdot (-2\cdot x)+\left(\frac{1}{2}\cdot x - \frac{1}{4} \right)\cdot (6)$     Distributive property
3. $\left(\frac{5}{2}\cdot x^{5}-\frac{5}{4}\cdot x^{4}\right)+\left(-x^{2}+\frac{1}{2}\cdot x \right)+\left(3\cdot x-\frac{3}{2} \right)$     Distributive property/$x^{m}\cdot x^{n} = x^{m+n}$
4. $\frac{5}{2}\cdot x^{5}-\frac{5}{4}\cdot x^{4}-x^{2}+\frac{7}{2}\cdot x-\frac{3}{2}$     Associative, distributive and commutative properties/Result

The product of $\frac{1}{2}\cdot x - \frac{1}{4}$ and $5\cdot x^{2}-2\cdot x + 6$ is $\frac{5}{2}\cdot x^{5}-\frac{5}{4}\cdot x^{4}-x^{2}+\frac{7}{2}\cdot x-\frac{3}{2}$. $\blacksquare$

To learn more on polynomials, we kindly invite to check this verified question: https://brainly.com/question/17822016