Washington (CNN) A majority of Americans say relations between blacks and whites in the US have worsened under President Barack Obama, according to a new CNN/ORC poll .

The survey results come as an increasing number say discrimination against blacks is a very serious problem and concerns about bias in the criminal justice system remain widespread.

Overall, 54% say relations between blacks and whites have gotten worse since Obama became president, including 57% of whites and 40% of blacks. That's up sharply compared with last June, when 43% said things had gotten worse shortly after a racially motivated shooting at a black church in Charleston, South Carolina.

A similar share, 52%, say the criminal justice system in the US favors whites over blacks, while 36% say it treats blacks and whites equally, down 6 points since February 2015.

About three-quarters of blacks and one-half of whites feel the system favors whites. Whites are sharply split by education level on this question, with 62% of whites who hold college degrees saying the system favors whites over blacks vs. just 42% among whites who do not have college degrees. And urbanites are more apt to say the system is discriminatory than are suburban or rural Americans: 62% who live in urban areas say so, above the 53% of suburbanites and 38% of rural residents who say the same.

The share describing discrimination against blacks generally as a very serious problem has inched up from 37% last June to 42% now, a high point during Obama's time in office. Much of the recent shift comes among whites, 34% of whom now see it as a very serious problem, up from 28% last June. Among blacks, that figure has softened somewhat and remains starkly higher than the share among whites, 80% said it was a very serious problem in June 2015 and 73% say so now.

Few Americans see their local police as prejudiced against blacks, but there are sharp differences by race and between those who live in urban areas vs. rural ones.

Nationwide, 18% say at least some of their local police are prejudiced against blacks. That climbs to 43% among blacks, and dips to 13% among whites. Among urbanites, 25% see at least some of their local police as prejudiced vs. just 10% among those who live in rural areas.

In general, Americans have broadly favorable opinions of their local police, 86% view them favorably, 10% unfavorably, with blacks and other non-whites a bit less likely to hold a favorable opinion (91% of whites have a favorable view, 77% of all non-whites, and 69% of blacks feel the same).

The Black Lives Matter movement, which emerged after several incidents in the last few years in which unarmed black people were harmed or killed at the hands of police, holds a narrowly net-positive favorability rating: 44% view it favorably, 40% unfavorably, and 16% have never heard of the movement or don't know how they feel about it.

Awareness of the movement has grown since last summer, when 33% had no opinion or had never heard of it.

Among blacks, 78% have a favorable view of the movement, higher than the 38% of whites who feel the same. There's a sharp divide among whites by education level in perceptions of Black Lives Matter, with 50% of whites who hold college degrees saying they see it favorably vs. 32% among whites without college degrees.

Two-thirds of adults say that peaceful protests are justified in response to those recent incidents where blacks where harmed or killed by police, and 14% overall, including about one-quarter of blacks, consider violent protests in response justified.

In findings from the same poll released earlier this week, voters were about evenly split between Hillary Clinton and Donald Trump on who would do a better job handling the criminal justice system. About 48% said Clinton would do the better job and 46% said they preferred Trump's approach.

Among whites, Trump held an edge 49% to 42% while blacks favored Clinton's approach 85% to 11%. Overall, Clinton held a wide advantage on improving the lives of racial minorities, 60% to 32% who thought Trump would be better on that score.

The CNN/ORC Poll was conducted by telephone September 28 through October 2 among a random national sample of 1,501 adults, including 1,086 non-Hispanic whites and 140 non-Hispanic blacks. The margin of sampling error is plus or minus 2.5 percentage points for results among the full sample, 3 points for results among whites and 8.5 points for results among blacks.